Let $G$ be a finite $p$-group and $N$ a maximal subgroup (so $G/N$ has order $p$) such that $Z(N) \leq Z(G)$. III.19.2 in Huppert's Book "Endliche Gruppen I" says that there exists a non-inner automorphism of $G$ of order $p$.
In the proof of Hilfssatz III.19.2, Huppert starts with a homomorphism $\tau: G\to Z(N)$ which is non-trivial and has $\ker\tau = N$. He makes no statement on why such a homomorphism exist.
How does one prove that such a homomorphism exists? Can one even explicitly describe such a homomorphism? Thanks in advance!
So, to try to make such a morphism (somewhat) explicit, we have little choice but to start with the canonical projection $\pi:G\to G/N\cong \Bbb Z_p.$
Then since $N$ is a $p$-group, it has non-trivial center $Z(N).$ So it's got an element of order $p$, hence $\Bbb Z_p$ embedds in $\Bbb Z(N).$ Let $i:\Bbb Z_p\to Z(N)$ be an embedding.
Finally, just compose: $$\tau :=i\circ \pi.$$