Two famous classes of multilinear functions are the symmetric and alternating multilinear functions, which satisfy for each $\sigma \in S_n$, $\sigma f = f$ and $\sigma f = \text{sgn}\sigma f$, respectively.
Both of these conditions can be formulated in terms of group actions; if $S_n$ acts on the space of $n$-linear functions, we can take quotients of $S_n$ to form another action; thus the symmetric functions are those multilinear functions on which the permutation action of $S_n$ is the same as the trivial action, and the alternating functions are those multilinear functions on which the permutation action of $S_n$ is the same as the sign action.
For almost every $n$, the symmetric group has only three quotients: $S_n$ itself, $C_2$ (which gives the sign action), and $e$ (which gives the trivial action). However, for $n=4$ we have that $S_3$ is a quotient of $S_4$, as the Klein 4-group is normal in $S_4$.
Here is my question: is there a way that $S_3$ acts on the set of $4$-linear functions (in a natural way, in the same way that $C_2$ and $e$ do), and is there a name for the collection of functions whose image under this action is the same as that under the permutation action?
It will be a little simpler to talk instead about elements of the tensor product $V^{\otimes n}$. These can be thought of as multilinear functions on the dual $V^{\ast}$ if you like. In the tensor product we can distinguish two natural subspaces, the symmetric tensors (those fixed under the action of $S_n$) and the alternating tensors (those which transform under the sign representation of $S_n$), and it's natural to ask how to generalize this.
The answer is as follows. For any irreducible representation of $S_n$, we can ask for the corresponding isotypic component of $V^{\otimes n}$, namely the subspace consisting of copies of that irreducible representation, or said another way the subspace consisting of elements which "transform under" that representation. The irreducible representations $M_{\lambda}$ of $S_n$ can be labeled by partitions $\lambda$, and the corresponding isotypic component of $V^{\otimes n}$ can canonically be identified with the tensor product (of vector spaces) of $M_{\lambda}$ and the Schur functor $S^{\lambda}(V)$. It's an aspect of Schur-Weyl duality that these are irreducible representations of $GL(V)$ (under some hypotheses on the ground field, I'm not sure what exactly we need but characteristic $0$ at least to be safe).
Your question is slightly different but the point of talking about Schur functors is that it cleanly generalizes to every symmetric group and does not require talking about quotients, which are in short supply. You can say the following: if $\varphi : S_4 \to S_3$ is the usual quotient, you can consider the subspace of $V^{\otimes 4}$ which is fixed by $\text{ker}(\varphi)$. This is, more or less by definition, the maximal subspace on which the action of $S_4$ factors through $\varphi$. As a sum of Schur functors it will correspond to those irreducible representations of $S_4$ which factor through $\varphi$, of which there are $3$ (out of $5$), namely the trivial representation, the sign representation, and the $2$-dimensional irreducible representation of $S_3$. The first two give us the symmetric and alternating tensors and the third gives (two copies of) some other Schur functor.