I am interested in studying automorphisms of a $p$-group, or at least the highest power of $p$ dividing the order of the automorphism group, and feel like studying homomorphism $G\to\text{Aut}(G)$ might be useful. When are there homomorphisms $G\to\text{Aut}(G)$ other than the obvious one $G\to\text{Inn}(G)$?
Another possible source of information is to study the representations $$\rho_K:\text{Aut}(G)\to GL(\mathbb{C}K)$$ sending an automorphism to its restriction to a characteristic subgroup $K$ (extended linearly to $\mathbb{C}K$). The character counts the number of fixed points of each automorphism, a subgroup of $K$, and is therefore a power of $p$, so it may (or may not) be useful. Has anyone seen this route pursued before?