Let $G$ be a finitely presented groups defined by
$$G=\{x_1,\ldots,x_n\mid R_1(x_1,\ldots,x_n)=\cdots=R_m(x_1,\ldots,x_n)=1 \}.$$
Let this presentation be denoted by $P$.
Let $S_n$ be the symmetric group on $\{1,\cdots,n\}$. Define
$G_*(P)=\{x_1,\ldots,x_n\mid \bigcup_{\sigma\in S_n} R_1(x_{\sigma(1)},\ldots,x_{\sigma(n)})=\cdots= R_m(x_{\sigma(1)},\ldots,x_{\sigma(n)})=1\}$.
For example, if $G=\{x,y\mid x^2=y^4=xyxy=1\}$, then $$G_*(P)=\{x,y\mid x^2=y^4=xyxy=x^4=y^2=yxyx=1\},$$ which is isomorphic to the Klein four group.
My question:
It seems natural that people has studied this before. What is the name for this "symmetrising the relations" process and where can I find the references?