  We consider the general circumstance of an Azumaya algebra $A$ of degree $n$
over a locally ringed topos $(\mathbf{X}, {\mathcal{O}}_{\mathbf{ X}})$ where
the latter carries a (possibly trivial) involution, denoted $\lambda$. This
generalizes the usual notion of involutions of Azumaya algebras over schemes
with involution, which in turn generalizes the notion of involutions of central
simple algebras. We provide a criterion to determine whether two Azumaya
algebras with involutions extending $\lambda$ are locally isomorphic, describe
the equivalence classes obtained by this relation, and settle the question of
when an Azumaya algebra $A$ is Brauer equivalent to an algebra carrying an
involution extending $\lambda$, by giving a cohomological condition. We remark
that these results are novel even in the case of schemes, since we allow
ramified, nontrivial involutions of the base object. We observe that, if the
cohomological condition is satisfied, then $A$ is Brauer equivalent to an
Azumaya algebra of degree $2n$ carrying an involution. By comparison with the
case of topological spaces, we show that the integer $2n$ is minimal, even in
the case of a nonsingular affine variety $X$ with a fixed-point free
involution. As an incidental step, we show that if $R$ is a commutative ring
with involution for which the fixed ring $S$ is local, then either $R$ is local
or $R/S$ is a quadratic \'etale extension of rings.
