  Let $(\mathcal{K},\mathcal{O},k)$ be a $p$-modular system with $k$
algebraically closed, let $b$ be a block of the normal subgroup $H$ of $G$
having defect pointed group $Q_\delta$ in $H$ and $P_\gamma$ in $G$, and
consider the block extension $b\mathcal{O}G$. One may attach to $b$ an extended
local category $\mathcal{E}_{(b,H,G)}$, a group extension $L$ of $Z(Q)$ by
$N_G(Q_\delta)/C_H(Q)$ having $P$ as a Sylow $p$-subgroup, and a cohomology
class $[\alpha]\in H^2(N_G(Q_\delta)/QC_H(Q),k^\times)$. We prove that these
objects are invariant under the $G/H$-graded basic Morita equivalences. Along
the way, we give alternative proofs of the results of K\"ulshammer and Puig
(1990), Puig and Zhou (2012) on extensions of nilpotent blocks. We also deduce
by our methods a result of Zhou (2016) on $p'$-extensions of inertial blocks.
