This post was inspired by this answer of Dave Penneys.

In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as the $W^*$-morphisms $\phi: M \to M'$ with $\phi (N) \subset N'$.

Unfortunately, through this definition, the category of finite group is $\underline{not}$ a (natural) subcategory of the category of subfactors.
In fact, let $G$ and $G'$ be finite groups and $f: G \to G'$ be a surjective group-morphism, then in general, $f$ does $\underline{not}$ extend into a (usual) subfactor-morphism of $(R \subset R \rtimes G)$ to $(R \subset R \rtimes G')$.

Here is the explanation in the answer of Dave:
A II$_1$-factor is algebraically simple, so each morphism of II$_1$-factors is either injective or zero.
Thus every non-zero morphism is an isomorphism onto its image.
I don't think the canonical surjection $G\to G'=G/\ker(f)$ actually gives you a map of factors $R\rtimes G\to R\rtimes G'$. In particular, if we denote the implementing unitaries as $u_g$ for $g\in G$, the map $u_g\mapsto u_{g\ker(f)}$ does not extend to a non-zero map of II$_1$-factors if $\ker(f)$ is non-trivial. The element $u_g-u_{g'}$ would map to zero if $g,g'\in \ker(f)$, and a non-trivial map of II$_1$-factors must be injective.

Question: Is there an $\underline{other}$ (natural) definition of subfactor-morphisms such that the category of finite groups is a (natural) subcategory of this "new" category of subfactors ?

This post was inspired by this answer of Dave Penneys.

In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as the $W^*$-morphisms $\phi: M \to M'$ with $\phi (N) \subset N'$.

Unfortunately, through this definition, the category of finite group is $\underline{not}$ a (natural) subcategory of the category of subfactors.
In fact, let $G$ and $G'$ be finite groups and $f: G \to G'$ be a surjective group-morphism, then in general, $f$ does $\underline{not}$ extend into a (usual) subfactor-morphism of $(R \subset R \rtimes G)$ to $(R \subset R \rtimes G')$.

Here is the explanation in the answer of Dave:
A II$_1$-factor is algebraically simple, so each morphism of II$_1$-factors is either injective or zero.
Thus every non-zero morphism is an isomorphism onto its image.
I don't think the canonical surjection $G\to G'=G/\ker(f)$ actually gives you a map of factors $R\rtimes G\to R\rtimes G'$. In particular, if we denote the implementing unitaries as $u_g$ for $g\in G$, the map $u_g\mapsto u_{g\ker(f)}$ does not extend to a non-zero map of II$_1$-factors if $\ker(f)$ is non-trivial. The element $u_g-u_{g'}$ would map to zero if $g,g'\in \ker(f)$, and a non-trivial map of II$_1$-factors must be injective.

Question: Is there an $\underline{other}$ (natural) definition of subfactor-morphisms such that the category of finite groups is a (natural) subcategory of this "new" category of subfactors ?