Number of subgroups of a $p$-group of index $p^k$

Question

Let $p$ be a prime, let $n$ and $k$ be positive integers and let $G$ be a group of order $p^n$. Further, let $a_{p^k}$ denote the number of subgroups of $G$ of index $p^k$.

If $a_{p^k}$ is greater than 1 and not congruent to $p+1$ modulo $p^2$ -- does it follow that $p = 2$ and $G$ is either a dihedral group, a quasidihedral group or a generalized quaternion group?

Answer 1

It seems that the $p>2$ part of this was proved in

Kulakoff, A., Über die Anzahl der eigentlichen Untergruppen und der Elemente von gegebener Ordnung in $p$-Gruppen., Math. Ann. 104, 778-793 (1931). ZBL57.0146.03.

and the $p=2$ part in

Easterfield, T. E., An extension of a theorem of Kulakoff, Proc. Camb. Philos. Soc. 34, 316-320 (1938). ZBL0019.10802.