Automorphisms of finite $p$-groups admitting a partition

For a finite $p$-group $P$, the following three conditions are equivalent: (a) to have a (proper) partition, that is, to be the union of some proper subgroups with trivial pairwise intersections; (b) to have a proper subgroup all elements outside which have order $p$; (c) to be a semidirect product $P=P_1\rtimes\langle\varphi\rangle$, where $P_1$ is a subgroup of index $p$ and $\varphi$ is its splitting automorphism of order $p$. It is proved that if a finite $p$-group $P$ with a partition admits a soluble automorphism group $A$ of coprime order such that the fixed-point subgroup $C_P(A)$ is soluble of derived length $d$, then $P$ has a maximal subgroup that is nilpotent of class bounded in terms of $p,d$, and $|A|$. The proof is based on a similar result derived by the author and P. V. Shumyatsky for the case where $P$ has exponent $p$ and on the method of elimination of automorphisms by nilpotency, which was earlier developed by the author, in particular, for studying finite $p$-groups with a partition. It is also proved that if a finite $p$-group $P$ with a partition admits an automorphism group $A$ that acts faithfully on $P/H_p(P)$, then the exponent of $P$ is bounded in terms of the exponent of $C_P(A)$. The proof of this result has its basis in the author's positive solution of an analog of the restricted Burnside problem for finite $p$-groups with a splitting automorphism of order $p$. The results mentioned yield corollaries for finite groups admitting a Frobenius group of automorphisms whose kernel is generated by a splitting automorphism of prime order.