Center not is fully invariant in class two p-group

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Statement

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

Related facts

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 center 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.