(Essentially, Burnside) If $H$ is a $p$-group containing a nonabelian characteristic subgroup with cyclic center, then there is no $p$-group $G$ such that $H$ is a $G$-invariant subgroup of $\Phi(G)$. In particular, $H\ne G'$, $H\ne \Phi(G)$.

(Essentially, Burnside) If $H$ is a $p$-group containing a nonabelian characteristic subgroup with cyclic center, then there is no $p$-group $G$ such that $H$ is a $G$-invariant subgroup of $\Phi(G)$. In particular, $H\ne G'$, $H\ne \Phi(G)$. Next, if a two-generator $H$ is $G$-invariant subgroup of $\Phi(G)$, then $H$ is metacyclic.

(Essentially, Burnside) If H is s p-group containing a nonabelian characteristic subgroup with cyclic center, then there is no p-group G such that H is a G-invariant subgroup of Phi(G). In particular, H\ne G', H\ne Phi(G).

(Essentially, Burnside) If $H$ is a $p$-group containing a nonabelian characteristic subgroup with cyclic center, then there is no $p$-group $G$ such that $H$ is a $G$-invariant subgroup of $\Phi(G)$. In particular, $H\ne G'$, $H\ne \Phi(G)$.