  We define equivariantization of tensor categories under tensor group scheme
actions and give necessary and sufficient conditions for an exact sequence of
tensor categories to be an equivariantization under a finite group or finite
group scheme action. We introduce the notion of central exact sequence of
tensor categories and use it in order to present an alternative formulation of
some known characterizations of equivariantizations for fusion categories, and
to extend these characterizations to equivariantizations of finite tensor
categories under finite group scheme actions. In particular, we obtain a simple
characterization of equivariantizations under actions of finite abelian groups.
As an application, we show that if $\C$ is a fusion category and $F: \C \to \D$
is a dominant tensor functor of Frobenius-Perron index $p$, then $F$ is an
equivariantization if $p=2$, or if $\C$ is weakly integral and $p$ is the
smallest prime factor of $\FPdim \C$.
