Socle not is fully invariant in class two p-group

Statement

Let $p$ be a prime number. There exists a p-group $G$ of class two whose Socle (?) is not a Fully invariant subgroup (?).

Related facts

• Center not is fully invariant
• Characteristic not implies fully invariant in odd-order class two p-group
• Center not is fully invariant in class two p-group
• Characteristic not implies fully invariant in class three maximal class p-group

Proof

Let $p$ be a prime. Let $P$ be any non-abelian group of order $p^{3}$ with center $Z$ (if $p=2$, choose $P$ to be dihedral group:D8. Otherwise there are two possibilities for $P$: a group of prime-square exponent, and a group of prime exponent). In all these groups, there is an element $x$ of order $p$ outside $Z$.

Define $G=P\times C$ where $C$ is the cyclic group of order $p$ with generator $y$. The socle of $G$ is the subgroup $H=Z\times C$.

Then $H$ is not fully invariant in $G$: Consider the retraction with kernel $P\times \{e\}$ and with image generated by the element $(x,y)$. This is an endomorphism of $G$, but it does not send $H$ to itself, since the element $(e,y)$ gets sent to $(x,y)$, which is outside $H$.