Below is part of an exercise in Section 6.1 of Abstract Algebra: Dummit and Foote.
26. (e) Let $p$ be a prime. Let $P$ be a finite $p$-group. Let $\sigma$ be any automorphism of $P$ of prime order $q$ with $q\neq p$. Show that if $\sigma$ fixes the coset $xΦ(P)$ then $\sigma$ fixes some element of this coset (Hint: note that since $Φ(P)$ is characteristic in $P$ every automorphism of $P$ induces an automorphism of $P/Φ(P)$).
What actual use is of the hint? At first glance I thought that it is of no use and this is my thought process: $(*)-$ Suppose $\sigma$ fixes $xΦ(P)$, a coset of order $p^a$ for some $a$. Since $|\sigma|=q$, its cycle decomposition as a permutation on elemenets of $P$ consists of $1$- and $q$-cycles only, i.e. an element is either fixed by $\sigma$, or is part of a $q$-cycle. So, we know that $\sigma$ must have a fixed point in $xΦ(P)$, since $q\nmid p^a$ implies there must be an element of $xΦ(P)$ not being part of any $q$-cycle. $-(*)$
Then I realized that this goes very wrong. I must be missing something since I didn't even use the hint! I'm guessing that the $\sigma$ that 'fixes the coset $xΦ(P)$' and the $\sigma$ that 'fixes some element of this coset' are actually two different things! So let's rename the former one, the one that 'fixes the coset $xΦ(P)$', as $\sigma_{ind}$. This may be what the authors actually have in mind: $$\sigma_{ind}:P/Φ(P)\rightarrow P/Φ(P)$$$$\sigma_{ind}(yΦ(P))=\sigma(y)Φ(P)$$ for all $y\in P$. $\sigma_{ind}\in\operatorname{Aut}(P/Φ(P))$ is said to be 'induced' by $\sigma$. It's a trivial task to check that $\sigma_{ind}$ is indeed an automorphism. The fact that $\sigma_{ind}$ is well-defined, i.e. the choice of coset representatives does not matter, results from $Φ(P)$ being a characteristic subgroup of $P$!
What's more: I found that if $\sigma_{ind}$ fixes the coset $xΦ(P)$ as an element of $P/Φ(P)$, then $\sigma$ also fixes $xΦ(P)$! Now $(*)$ finishes the proof.
Am I on the right track? Or do the authors actually mean another thing?
P.S. What's special about $Φ(P)$ here? Can it not be replaced by an arbitrary characteristic subgroup of $P$?