  A family of symmetric functions $\tilde{s}_\lambda$ was introduced in [OZ],
and independently in [AS]. The $\tilde{s}_\lambda$ encode many stability
properties of representations of symmetric groups (e.g. when multiplied, the
structure constants are reduced Kronecker coefficients). We show that the
structure constants for the Kronecker comultiplication $\Delta^*$ are
multiplicities for the restriction of irreducible representations from $S_{mn}$
to $S_m \times S_n$ (provided $m$ and $n$ are sufficiently large), and use the
structure of $\tilde{s}_\lambda$ to demonstrate two-row stability properties of
these restriction multiplicities.
