Let $\gamma\in S_{n+1}$ act naturally on $\Delta^n$, then for some $\sigma:\Delta^n\to X$, $\sigma\circ\gamma=\mathrm{sgn}(\gamma)\sigma$ in homology

Question

Suppose we are given a topological space $X$, and a singular simplex $\sigma:\Delta^n\to X$, where $\Delta^n$ is the convex hull of $e_0,\ldots,e_n\in\mathbb R^{n+1}$. For any permutation $\gamma\in S_{n+1}$, we can define $\gamma:\Delta^n\to\Delta^n$ by $$ \gamma\left(\sum_{i=0}^n \alpha_i e_i\right)=\sum_{i=0}^n \alpha_i e_{\gamma(i)}. $$ Intuitively I would expect that we have $\sigma\circ\gamma\sim\operatorname{sgn}(\gamma)\sigma$ in homology, i.e. $\sigma\circ\gamma-\operatorname{sgn}(\gamma)\sigma$ is a boundary. Clearly we can reduce to the case where $\gamma$ is a swap, but I've struggled to find a clear proof of this statement even in this case, since I don't see such an obvious candidate for a $\tau\in C_{n+1}(X)$ such that $d\tau=\sigma\circ\gamma-\operatorname{sgn}(\gamma)\sigma$.