necessary (cf. Theorem \[thm-commeq\]). The extension groups of objects in $\QMod_\mfb B$ can be obtained through the extension groups of the corresponding objects in $\QMod_\mfr R$ (cf. Proposition \[prop-qext\]), and thus we establish connections among extension groups of objects in $\QMod_\mfa A$, $\QMod_\mfr R$ and $\QMod_\mfb B$ respectively.

Throughout the paper, $\kk$ is a field. All the algebras and modules are assumed to be $\kk$-vector space. Unadorned $\otimes$ means $\otimes_{\kk}$. All the finite groups considered in the paper are nontrivial.

Radicals of group actions {#sec1}
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Let $R$ be a noetherian algebra. Let $G$ be a finite group which acts on $R$ by automorphisms so that $R$ is a left $G$-module algebra. The $G$-action determines an ideal of $R$.

\[def-pelmt\] We say that two sequences $(a_1,\dots,a_n)$ and $(b_1,\dots,b_n)$ of elements of $R$ are [*pertinent under the $G$-action*]{}, if $\sum_{i=1}^na_i(g\cdot b_i)=0$ for all $1\neq g\in G$. We write $(a_1,\dots,a_n)\overset{G}\sim(b_1,\dots,b_n)$ for pertinent sequences.

\[example1\] (i) Let $R=\kk[x,y]$. Let $\sigma$ be the automorphism of $A$ defined by $\sigma(x)=y$ and $\sigma(y)=x$. Let $G=\{1,g\}$. Then the sequences $(x,y)\overset{G}\sim(x,-y)$.

\(ii) Let $R=\kk_{-1}[x,y]$ be the skew symmetric algebra. Let $\sigma$ be the automorphism of $A$ defined by $\sigma(x)=y$ and $\sigma(y)=x$. Let $G=\{1,g\}$. Then the sequences $(x,y)\overset{g}\sim(x,y)$.

For 