Given two finite groups $G,H$ with a homomorphism $\phi:G\to H$ the homomorphism can be injective, surjective or neither. This subdivides the morphisms in the category of finite groups $\mathrm{FinGr}$ effectively into three disjoint parts.
My question: Is there a category theoretical construction which can help us to characterize how $\mathrm{Hom}(G,H)$ splits with respect to this property of its arrows?
The most important for me would be to preserve the doamin, so anything that works with $\mathrm{Hom}(G,-)$ is also ok. My approach was first to consider the under category of any finite group $G$ but I always seemed to lose the information when doing anything non trivial. The second attempt was then based on the arrow category which was a bit more fruitful since compositions of injective/surjective maps preserve these properties but now I lose the reference to the fixed domain $G$ which I can preserve in the first attempt using the under category construction.
Does anyone have an idea here?
EDIT: I identify isomorphic groups, so the homomorphisms cannot be bijective.
