Sets of $H$-fixed points are absolute extensors

The necessity part of a theorem of Jaworowski on the extension of periodic homeomorphisms is strengthened (RZhMat., 1973, 2A433). Let $\mathscr B$ $(\mathscr M)$ be the class of all compact Hausdorff spaces (metrizable spaces), and let $\mathscr B(G)$ ($\mathscr M(G)$) be the class of all compact Hausdorff spaces (metrizable spaces) considered with all possible actions of a topological group $G$.
Theorems B and M. {\it If a topological space $Y$ on which a group $G\in\mathscr B$ $(G\in\mathscr M)$ acts is an extensor of $\mathscr B(G)$ $(\mathscr M(G))$, then for every closed subgroup $H$ of $G$ the set $Y[H]=\{y\in Y\mid hy=y\ \forall\,h\in H\}$ of all "$H$-fixed points" is an extensor of the class $\mathscr B(\mathscr M)$.}
These theorems are also valid for the neighborhood case under the additional condition that for mappings $f\colon A\to Y[H]$ extendible to $X$ the dimension $\dim(X\setminus A)\leqslant n+1$, and for equivariant mappings $g\colon B\to Y$ extendible to $X$ the dimension $\dim(X\setminus B)\leqslant n+1+\dim G$.
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