  For a fixed finite group $Q$ and semi-simple finite dimensional algebra $S$,
we examine an equivalence between strongly $Q$-graded algebras (extensions)
with identity component $S$ and $S^1$-gerbes on action groupoids of $Q$ on the
set of isomorphism classes of simple objects of the category of $S$-modules.
This clarifies the nature of the map considered in arXiv:1312.7316. Motivated
by this and arXiv:0909.3140(2) we suggest and study a notion of extensions
suitable to the case when $S$ is replaced by a Hopf algebra, in the sense that
there is a bijection between extensions with "fiber" $H$ and $H^*$. In
particular we focus on the case of $H$ equal to the group algebra of a finite
group. When $K$ is abelian, the answer is particularly symmetric as duality of
Hopf algebras does not take us outside of the category of groups.
