  We show that, given an almost-source algebra $A$ of a $p$-block of a finite
group $G$, then the unit group of $A$ contains a basis stabilized by the left
and right multiplicative action of the defect group if and only if, in a sense
to be made precise, certain relative multiplicities of local pointed groups are
invariant with respect to the fusion system. We also show that, when $G$ is
$p$-solvable, those two equivalent conditions hold for some almost-source
algebra of the given $p$-block. One motive lies in the fact that, by a theorem
of Linckelmann, if the two equivalent conditions hold for $A$, then any stable
basis for $A$ is semicharacteristic for the fusion system.
