This is a refinement of a question asked on MSE.

Let $G$ be a finite group and let $V$ be a finite-dimensional faithful complex representation of $G$. Consider $V$ as an affine complex variety. In general (though not always), the image of the origin in the quotient variety $V/G$ is singular. By the "singularity type" of the pair $G,V$ I refer to the isomorphism class of the ring $\widehat{\mathcal{O}_{V/G,0}}$, the completion of the local ring at this point.

At broadest (soft-question) level, what I want to know is, "how fine an invariant of the pair $G,V$ is the singularity type?" But let me narrow the scope in order to be able to ask something precise:

Suppose $G$ is nonabelian, $V$ has no one-dimensional subrepresentations, and $V/G$ is singular at the image of the origin. Further suppose $A$ is a finite abelian group and $W$ is a faithful representation of $A$. Is it possible for $G,V$ to have the same singularity type as $A,W$?

Comments: (1) The answer is certainly "yes" without the stipulation that $V/G$ be singular at the origin; both abelian and nonabelian groups can have smooth quotients, by Chevalley-Shephard-Todd since both abelian and nonabelian groups can be reflection groups. (2) The stipulation "$V$ has no one-dimensional subrepresentations" is there to rule out a match resulting from $G,V$'s singularity "actually" coming from $G$'s abelianization. This can happen for example if $V$ decomposes as a sum $W\oplus L$ with $W$ such that $W/G$ is smooth and $L$ is a sum of one-dimensional representations.

This is a refinement of a question asked on MSE.

Let $G$ be a finite group and let $V$ be a finite-dimensional faithful complex representation of $G$. Consider $V$ as an affine complex variety. In general (though not always), the image of the origin in the quotient variety $V/G$ is singular. By the "singularity type" of the pair $G,V$ I refer to the isomorphism class of the ring $\widehat{\mathcal{O}_{V/G,0}}$, the completion of the local ring at this point.

At broadest (soft-question) level, what I want to know is, "how fine an invariant of the pair $G,V$ is the singularity type?" But let me narrow the scope in order to be able to ask something precise:

Suppose $G$ is nonabelian, $V$ has no one-dimensional subrepresentations, and $V/G$ is singular at the image of the origin. Further suppose $A$ is a finite abelian group and $W$ is a faithful representation of $A$. Is it possible for $G,V$ to have the same singularity type as $A,W$?

Comments: (1) The answer is certainly "yes" without the stipulation that $V/G$ be singular at the origin; both abelian and nonabelian groups can have smooth quotients, by Chevalley-Shephard-Todd since both abelian and nonabelian groups can be reflection groups. (2) The stipulation "$V$ has no one-dimensional subrepresentations" is there to rule out a match resulting from $G,V$'s singularity "actually" coming from $G$'s abelianization. This can happen for example if $V$ decomposes as a sum $W\oplus L$ with $W$ such that $W/G$ is smooth and $L$ is a sum of one-dimensional representations.