  To a strongly $G$-graded algebra $A$ with $1$-component $B$ we associate the
group $\mathrm{Picent}^{\mathrm{gr}}(A)$ of isomorphism classes of invertible
$G$-graded $(A,A)$-bimodules over the centralizer of $B$ in $A$. Our main
result is a $\mathrm{Picent}$ version of the Beattie-del R\'{\i}o exact
sequence, involving Dade's group $G[B]$, which relates
$\mathrm{Picent}^{\mathrm{gr}}(A)$, $\mathrm{Picent}(B)$, and group cohomology.
