  Every finite non-nilpotent group can be extended by a term operation such
that solving equations in the resulting algebra is NP-complete and checking
identities is co-NP-complete. This result was firstly proven by Horv\'ath and
Szab\'o; the term constructed in their proof depends on the underlying group.
In this paper we provide a uniform term extension that induces hardness. In
doing so we also characterize a big class of solvable, non-nilpotent groups for
which extending by the commutator operation suffices.
