  We show that the $p$-power maps in the first Hochschild cohomology space of
finite-dimensional selfinjective algebras over a field of prime characteristic
$p$ commute with stable equivalences of Morita type on the subgroup of classes
represented by integrable derivations. We show, by giving an example, that the
$p$-power maps do not necessarily commute with arbitrary transfer maps in the
Hochschild cohomology of symmetric algebras.
