$\newcommand{\Z}{\mathbb Z}\newcommand{\F}{\mathbb F}$I would like to define a procedure which turns a finite $p$-group $G$ together with a field $K$ of characteristic $p$ (or even, in fact, any ring of characteristic $p$) into a group $G_K$ such that $G_K$ is obtained by "replacing all the $\Z/p\Z$-factors of $G$ by $K$". That is to say: unscrew the $p$-group until only cyclic factors are left, then base-change those cyclic factors, and screw them again following the original assembly process.
Ideally, I would like this procedure to satisfy the following properties:
- When $K = \mathbb F_p$, I simply get $G_K = G$, and more generally for $K =\mathbb F_q$ with $q=p^d$ I get a group $G_K$ of order $|G|^d$
- When $G = \mathbb Z/p\Z$, I get $G_K = K$ (see below for the abelian case)
- This is functorial in $K$, so any embedding of fields $K \hookrightarrow L$ gives an embedding $G_K \subseteq G_L$; in particular, this would apply to automorphisms $\sigma$ of $K$ and then I would expect the fixed subgroup of $G_K$ under the group automorphism induced by $\sigma$ to be $G_{K^\sigma}$
- For a fixed field $K$, the functor $G \to G_K$ is "exact" in the sense that for any exact sequence $1 \to A \to B \to C \to 1$ of $p$-groups I recover an exact sequence $1 \to A_K \to B_K \to C_K \to 1$ of $p$-groups
When $G$ is abelian (so a finite $\Z_p$-module), I know how to do this: I can simply consider $G \otimes_{\Z_p} W(K)$, or maybe I can write down an isomorphism $G \simeq \bigoplus \Z/p^{n_i}\Z$ and then have $G_K = \bigoplus W_{n_i}(K)$ (when $K$ is perfect these two options are equivalent).
Another (more general) case where I know how to deal with this is when $G$ has nilpotency class $<p$. In that case, the Lazard correspondence tells us that $G$ corresponds to some finite Lie $\Z_p$-algebra $\mathfrak g$, namely that $G$ can be recovered by equipping $\mathfrak g$ with the "Baker-Campbell-Hausdorff group law" $\circ$ such that $x \circ y = x + y + \frac12 [x,y] + \ldots$ where one truncates the Baker-Campbell-Hausdorff formula as soon as there are $p$-th commutators. But then one can bilinearly extend $\mathfrak g$ and its Lie bracket into a Lie $W(K)$-algebra, and equip that Lie algebra with its group law $\circ$ to obtain the group $G_K = (\mathfrak g \otimes_{\Z_p} W(K), \circ)$.
I do not know how this would go in more generality, but there are certainly instances where I know what I would expect, for example if $G$ is the group of unipotent upper-triangular matrices of size $n$ over $\F_p$, one would expect $G_K$ to be the group of unipotent upper-triangular matrices of size $n$ over $K$, etc. (typically for the group of $\mathbb F_p$-points of an unipotent algebraic group over $\mathbb F_p$) An approach that sounds promising, focusing just on the case of groups of nilpotency class $2$, given as $$ 1 \to Z \to G \to Q \to 1$$ with both $Z = Z(G)$ and $Q=G/Z(G)$ abelian. Then the extension $G$ is characterized by an element of $H^2(Q,Z)$, I know how I want to base-change both $Z$ and $Q$, so I would actually want a map $H^2(Q,Z) \to H^2(Q_K, Z_K)$. Shapiro's lemma comes close to giving me such a map, namely I get an isomorphism $H^2(Q,Z) \simeq H^2(Q_K, \mathrm{Coind}_Q^{Q_K} Z)$. However we do not have an isomorphism $\mathrm{Coind}_Q^{Q_K} Z \simeq Z_K$ unless $K$ is a finite field (in which case this does work!). This is all the more mysterious that I know the map $H^2(Q,Z) \to H^2(Q_K, Z_K)$ exists whenever $p>2$: take the corresponding extension $G$, turn it into a Lie algebra, base-change it, turn it back into a group, and take the corresponding element of $H^2$. It seems very mysterious to me that this kind of "Shapiro-like lemma" with the coinduced replaced by the tensor product would hold unless perhaps when $p=2$, and I do not understand the cohomological significance of the fact that it does hold when the Lazard correspondence applies.
Does anyone have any knowledge on previous work on this question? Do these constructions of "extensions of scalars of $p$-groups" have a common name? Are there obstructions, for certain finite $p$-groups, for an extension of scalars to exist?
EDIT: I'm mentioning another case I know how to deal with: assume that G has nilpotency class 2 and exponent p. Write $1 \to Z \to G \to Q \to 1$ with $Z=Z(G)$ and $Q=G/Z(G)$, and pick an $\F_p$-basis $b_1, \ldots, b_r$ of $Q$. For each basis element pick a lift $\tilde b_i \in G$. Then, choose the "almost morphic" (the single arbitrary choice for which we have to pay a price is some ordering of the basis elements) set-theoretic section $s \colon Q \to G$ defined by $$ \sum \lambda_i b_i \mapsto \tilde b_1^{\lambda_1} \cdots \tilde b_r^{\lambda_r} $$ this is well-defined because $\lambda_i$ is unique as an element of $\F_p$, and each $\tilde b_i$ has order $p$ (if we do not assume $G$ has exponent $p$ we have to deal with carrying in the exponents). Let $z_{i,j} = [\tilde b_i, \tilde b_j] \in Z$. Consider two elements $x = \sum \lambda_i b_i$ and $y = \sum \mu_i b_i$ of $Q$. We have $$ s(x)s(y) = \tilde b_1^{\lambda_1} \cdots \tilde b_r^{\lambda_r} \cdots \tilde b_1^{\mu_1} \cdots \tilde b_r^{\mu_r} = c(x,y) s(x+y) $$ where $c(x,y) = \sum_{i<j} \lambda_i \mu_j z_{i,j}$, so in this particular case we actually do find a bilinear $2$-cochain representing the class in $H^2(Q,Z)$, and then we can simply bilinearly extend this class by tensoring with $K$.
If we do not assume that $G$ has exponent $p$, it may happen that the order of $\tilde b_i$ is larger than that of $b_i$, and then we have to pick an even more uncanonical set-theoretic section, for example by picking the smallest integer as the exponent, and then we have to deal with carrying. I think base-changing should still be possible, by replacing the carrying cocycle by Witt polynomials.
I have to say I would like a cleaner way of constructing this extension, especially a way that makes functoriality clear, and ideally a way that generalizes to any $p$-group of any nilpotency class and any exponent. My general impression, right now, is that any $p$-group, say of order $p^d$, can be represented by having $d$ coordinates in $\F_p$, and the group law is addition "corrected" by some "strictly triangular mapping" (the correction at a given coordinate depends only of the preceding coordinates), where that correction is a mixture of multilinear maps and Witt polynomials. If that could somehow be made correct, then we could simply take the same description but allowing coordinates in a larger ring to obtain extensions of scalars.
