Set of homomorphisms form a group? Given vector spaces $V, W$ over field $F$, the set of all linear maps $V \to W$ forms a vector space over $F$ under pointwise addition.
Is there an analogue for groups?  Can the set of all homomorphisms from groups $G \to K$ be given a group structure?

If we assume that the target group $K$ is abelian, then the set of homomorphisms $\operatorname{Hom}(G, K)$ form an abelian group under the point-wise multiplication $$(\phi\psi)(g)=\phi(g)\psi(g)$$  with identity $1(g)=e_K$ (identity of $K$), and inversion $(\phi)^{-1}(g)=(\phi(g))^{-1}=\phi(g^{-1}).$
