Finite $p$-Groups with Automorphism of a Special Form

Research on finite solvable groups with $C$-closed invariant subgroups has given rise to groups structured as follows. Let $p,q_1,q_2,\dots,q_m$ be distinct primes, $n_i$ be the exponent of $p$ modulo $q_i$, and $n$ be the exponent of $p$ modulo $r=\prod\limits_{i=1}^m q_i$. Then $G=P\lambda\langle x\rangle$, where $P$ is a group and $Z(P)=P'=\prod\limits_{i=1}^{m}Z_i$; here, $Z_i$ and $P/Z(P)$ are elementary Abelian groups of respective orders $p^{n_i}$ and $p^n$, $|x|=r$, the element $x$ acts irreducibly on $P/Z(P)$ and on each of the subgroups $Z_i$, and $C_P(x^{q_i})=Z_i$. We state necessary and sufficient conditions for such groups to exist.