  The Schur functions, a basis for the symmetric polynomials (Sym), encode the
irreducible representations of the symmetric group, $\mathfrak{S}_n$, via the
Frobenius characteristic map. In 1996, Krob and Thibon defined a quasisymmetric
Frobenius map on the representations of $\mathcal{H}_n(0)$, mapping them to the
quasisymmetric functions (QSym). Despite the obvious inclusion of Sym in QSym
and the close relationship between $\mathfrak{S}_n$ and $\mathcal{H}_n(0)$,
there is no known direct link between these two Frobenius characteristic maps
and the related representations. We explore three specific situations in which
a deformation of an $\mathfrak{S}_n$ action results in a valid
$\mathcal{H}_n(0)$ action and gives a quasisymmetric Frobenius characteristic
that is equal to the symmetric Frobenius characteristic. We introduce the
concept of quasisymmetric compatibility, which formalizes a link between the
two maps, and we show it applies to all $\mathfrak{S}_n$-modules.
