A weakly equivariant Hopf algebra is a Hopf algebra A with an action of a finite group G up to inner automorphisms of A. We show that each weakly equivariant Hopf algebra can be replaced by a Morita equivalent algebra Astr with a strict action of G and with a coalgebra structure that leads to a tensor equivalent representation category. However, the coproduct of this strictification cannot, in general, be chosen to be unital, so that a strictification of the G-action can only be found on a \emph{weak} Hopf algebra Astr.