Let $X=S^2\setminus D$, for $D\subset S^2$ some finite set of points, say with $|D|=n\geq 1$. The category of locally constant sheaves of $\mathbb{C}$-vector spaces on $X$ (equivalently, complex representations of $\pi_1(X)=F_{n-1}$), $\text{LocSys}(X)$, has many natural autoequivalences.

For example, if $f: S^2\to S^2$ is a homeomorphism fixing $D$, then $\mathbb{V}\mapsto f^*\mathbb{V}$ is an autoequivalence of $\text{LocSys}(X)$. Two homeomorphisms fixing $D$, homotopic through homeomorphisms fixing $D$, induce naturally equivalent autoequivalences, so one obtains an action of the mapping class group $\text{Mod}(S^2, D)$ on $\text{LocSys}(X)$. Since $\text{Mod}(S^2, D)$ is the pure spherical braid group on $n$ strands, I'll denote it by $B_n$. My question is:

What are the auotequivalences of $\text{LocSys}(X)$ commuting (up to natural isomorphism) with the actions of $B_n$?

I know of three sources of examples. The first is invertible objects in $\text{LocSys}(X)$: if $\mathbb{L}$ is a rank one local system on $X$, the functor $\mathbb{V}\mapsto \mathbb{V}\otimes \mathbb{L}$ is $B_n$-equivariant. The second is the evident action of the group $\text{Aut}(\mathbb{C}/\mathbb{Q})$ on the structure constants of any given local system. Both of these operations preserve the rank of a local system.

The other is Katz's middle convolution operation, which does not.

Let $\Delta\subset X\times X$ be the diagonal, $\pi_1, \pi_2: X\times X\setminus \Delta\to X$ the two projections, $a: X\times X\setminus \Delta\to \mathbb{C}^\times$ the map $(x, y)\mapsto x-y$ (where here we choose an isomorphism $S^2\simeq \mathbb{CP}^1$ sending some point of $D$ to $\infty$ to coordinatize $S^2$), and $j: X\times X\setminus \Delta\hookrightarrow S^2\times X$ the evident inclusion. Then given a non-trivial rank one local system $\chi$ on $\mathbb{C}^\times$, the middle convolution $$MC_{\chi}(\mathbb{V}):=R^1\pi_{2\ast}j_{\ast}(\pi_1^{\ast}\mathbb{V}\otimes a^{\ast}\chi).$$ It is not obvious but Katz shows it is true that $MC_\chi$ is (up to equivalence) inverse to $MC_{\chi^{-1}}$. Moreover this operation (not entirely obviously) commutes with the action of $B_n$.

Of course one can compose these operations in various ways to get more autoequivalences commuting with the $B_n$-action.

Is that everything?

I'm also curious in the analogous question with $S^2\setminus D$ replaced by a surface of higher genus.

Let $X=S^2\setminus D$, for $D\subset S^2$ some finite set of points, say with $|D|=n\geq 1$. The category of locally constant sheaves of $\mathbb{C}$-vector spaces on $X$ (equivalently, complex representations of $\pi_1(X)=F_{n-1}$), $\text{LocSys}(X)$, has many natural autoequivalences.

For example, if $f: S^2\to S^2$ is a homeomorphism fixing $D$, then $\mathbb{V}\mapsto f^*\mathbb{V}$ is an autoequivalence of $\text{LocSys}(X)$. Two homeomorphisms fixing $D$, homotopic through homeomorphisms fixing $D$, induce naturally equivalent autoequivalences, so one obtains an action of the mapping class group $\text{Mod}(S^2, D)$ on $\text{LocSys}(X)$. Since $\text{Mod}(S^2, D)$ is the pure spherical braid group on $n$ strands, I'll denote it by $B_n$. My question is:

What are the auotequivalences of $\text{LocSys}(X)$ commuting (up to natural isomorphism) with the actions of $B_n$?

I know of three sources of examples. The first is invertible objects in $\text{LocSys}(X)$: if $\mathbb{L}$ is a rank one local system on $X$, the functor $\mathbb{V}\mapsto \mathbb{V}\otimes \mathbb{L}$ is $B_n$-equivariant. The second is the evident action of the group $\text{Aut}(\mathbb{C}/\mathbb{Q})$ on the structure constants of any given local system. Both of these operations preserve the rank of a local system.

The other is Katz's middle convolution operation, which does not.

Let $\Delta\subset X\times X$ be the diagonal, $\pi_1, \pi_2: X\times X\setminus \Delta\to X$ the two projections, $a: X\times X\setminus \Delta\to \mathbb{C}^\times$ the map $(x, y)\mapsto x-y$ (where here we choose an isomorphism $S^2\simeq \mathbb{CP}^1$ sending some point of $D$ to $\infty$ to coordinatize $S^2$), and $j: X\times X\setminus \Delta\hookrightarrow S^2\times X$ the evident inclusion. Then given a non-trivial rank one local system $\chi$ on $\mathbb{C}^\times$, the middle convolution $$MC_{\chi}(\mathbb{V}):=R^1\pi_{2\ast}j_{\ast}(\pi_1^{\ast}\mathbb{V}\otimes a^{\ast}\chi).$$ It is not obvious but Katz shows it is true that $MC_\chi$ is (up to equivalence) inverse to $MC_{\chi^{-1}}$. Moreover this operation (not entirely obviously) commutes with the action of $B_n$.

Of course one can compose these operations in various ways to get more autoequivalences commuting with the $B_n$-action.

Is that everything?

I'm also curious about the analogous question with $S^2\setminus D$ replaced by a surface of higher genus.