Category on a finite p-group

Definition
A category on a finite p-group is a category defined relative to a group of prime power order as follows. Let $$P$$ be a group of prime power order where the prime is $$p$$. The category $$\mathcal{F}$$ can be described as follows:


 * Its objects are all the subgroups of $$P$$
 * It is a subcategory of the category of all possible injective homomorphisms between these objects, under composition. In other words, for any two subgroups $$Q$$ and $$R$$, the homomorphism set $$\operatorname{Hom}_{\mathcal{F}}(Q,R)$$ is a subset of the set of all injective homomorphisms from $$Q$$ to $$R$$, and composition of morphisms is composition as injective homomorphisms.
 * It contains all inclusion maps. So, if $$Q \le R \le P$$, then the natural inclusion of $$Q$$ in $$R$$ is a member of $$\operatorname{Hom}_{\mathcal{F}}(Q,R)$$.
 * If $$\varphi:Q \to R$$ is a morphism of $$\mathcal{F}$$, the restriction with co-domain $$\varphi(Q)$$ is also a morphism of $$\mathcal{F}$$, and so is the inverse of that restriction. In particular, $$Q \cong \varphi(Q)$$.