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Let $p$ be a prime, $P$ a finite p-group and $\cal F$ a Frobenius$P$-category. In "Existence, uniqueness and functoriality of the perfectlocality over a Frobenius $P$-category", Algebra Colloquium, 23(2016) 541-622,we also claimed the uniqueness of the partial perfect locality $\cal L^{\frakX}$ over any up-closed set $\frak X$ of $\cal F$-selfcentralizing subgroups of$P$, but recently Bob Oliver exhibit some counter-examples, demanding somerevision of our arguments. In this Note we show that, up to replacing theperfect localities by the "extendable" perfect localities over any up-closedset $\frak X$ of $\cal F$-selfcentralizing subgroups of $P$, our arguments arecorrect, still proving the existence and the uniqueness of the perfect $\calF^{\rm sc}$-locality, since it is "extendable". We take advantage to simplifysome of our arguments.

A correction to the uniqueness of a partial perfect locality over a Frobenius P-category