For $r \in \mathbb{Z}_{>0}$, we say that a tower $1=G_0 \subseteq G_1 \subseteq \cdots$ of finite groups is an $r$-differential tower if for all $n$ the branching rules for restriction of irreducible (complex) representations from $G_n$ to $G_{n-1}$ are multiplicity free, and we have that $$ Res^{G_{n+1}}_{G_n} Ind_{G_{n}}^{G_{n+1}} - Ind_{G_{n-1}}^{G_n} Res^{G_{n}}_{G_{n-1}} = r \cdot id $$ viewed as a linear operator on the representation ring $R(G_n)$.
For example, the tower of symmetric groups $\mathfrak{S}_0 \subseteq \mathfrak{S}_1 \subseteq \mathfrak{S}_2 \subseteq \cdots$, and more generally the tower $(A \text{ wr } \mathfrak{S}_n)_{n \geq 0}$ of wreath products of a fixed abelian group $A$ and the symmetric groups give $r$-differential towers where $r=|A|$. This is a restatement of the fact that Young's lattice $Y$ and its powers $Y^r$ are $(r$-)differential posets.
Are there any other known examples of differential towers of groups? I would be interested in even a small modification of these families (involving, say, alternating groups instead of symmetric groups, or allowing the abelian group $A$ to depend on $n$).