The question is self-contained finite group theory but the motivation requires more background.

The finite groups I am interested in are the groups $Sp(n,F_3)$. For $n$ even these are the usual symplectic groups over the field with three elements. For $n$ odd these are the odd symplectic groups. These are a semi-direct product of a symplectic group with a Heisenberg group and we have a sequence of subgroups $Sp(n,F_3)\rightarrow Sp(n+1,F_3)$.

Consider one of these inclusions and look at induction/restriction of irreducible complex representations. My question is: take an irreducible representation of the subgroup and induce. Is this representation a direct sum of pair-wise non-isomorphic irreducible representations (sometimes this is called multiplicity free)?

I can prove this for $n$ even because the irreducible representations of $Sp(2m+1,F_3)$ can be constructed using Clifford theory (known to physicists as Mackey theory). I can use the computer for small $n$. So my question is really for $n$ odd.

I can give more information which hints at my interest. Define a sequence of groups $G(n)$ by a sequence of finite presentations so that we have surjective homomorphisms $B(n)\rightarrow G(n)$ where $B(n)$ is the usual braid group. Take generators $\sigma_1,\ldots ,\sigma_{n-1}$ and the Artin relations. In addition take $\sigma_i^3$ and for $n\ge 5$ take $(\sigma_1\sigma_2\sigma_3\sigma_4)^{10}=1$. Then we have $G(n)=Sp(n-1,F_3)$.

Then we also have $G(n)\times G(m)\rightarrow G(n+m)$ compatible with $B(n)\times B(m)\rightarrow B(n+m)$. This is similar to the symmetric groups.

I am aware that this question can be generalised. I have deliberately restricted to a simple example as I would like to have one case fully worked out before generalising. If you have a proof of the above and your proof generalises, that's different!