  A subset $X$ of a finite group $G$ is said to be prime-power-independent if
each element in $X$ has prime power order and there is no proper subset $Y$ of
$X$ with $\langle Y, \Phi(G)\rangle = \langle X, \Phi(G)\rangle$, where
$\Phi(G)$ is the Frattini subgroup of $G$. A group $G$ is $\mathcal{B}_{pp}$ if
all prime-power-independent generating sets for $G$ have the same cardinality.
We prove that, if $G$ is $\mathcal{B}_{pp}$, then $G$ is solvable. Pivoting on
some recent results of Krempa and Stocka, this yields a complete classification
of $\mathcal{B}_{pp}$-groups.
