or ${\partial\Delta}^n$, then ${\mathrm{Aut}}({\mathcal{M}}(K)) \cong {\mathrm{Aut}}(K)$ .

As in Proposition \[prop: autK subgroup\], define a function $\phi\colon \text{Aut}(K) \to \text{Aut}(\mathcal{M}(K))$ that sends each simplicial automorphism $f \in \text{Aut}(K)$ to its induced simplicial automorphism $f_{\ast}\in \text{Aut}(\mathcal{M}(K))$. We know that $\phi$ is an injective homomorphism by Proposition \[prop: autK subgroup\].

We now show that there is a surjection onto ${\mathrm{Aut}}({\mathcal{M}}(K))$. If so, since ${\mathcal{M}}(K)$ and $K$ are finite, this implies that $\phi$ is an isomorphism. Let $F\in {\mathrm{Aut}}({\mathcal{M}}(K)).$ If $K$ is a 1-dimensional simplicial complex (i.e., a graph) other than $C_n$, then $F$ is induced by a simplicial isomorphism $f\colon K\to K$ by [@CM-17 Theorem 3.5]. Thus the result for $K$ a graph other than $C_n$.

Now suppose that $K\neq {\partial\Delta}^n$ is a simplicial complex of dimension greater than or equal to $2$, and let $F\in {\mathrm{Aut}}({\mathcal{M}}(K)).$ In the proof of Theorem A [@CM-17], Capitelli and Minian construct a simplicial isomorphism $f\colon K \to K$ that induces the given $F\colon {\mathcal{M}}(K)\to {\mathcal{M}}(K).$ Their construction of $f$ relies on a condition that is not satisfied for $K={\partial\Delta}^n$ according to the contrapositive of [@CM-17 Theorem 4.2]. Thus the result for $K\neq {\partial\Delta}^n$ a simplicial complex of dimension greater than $2$.

The Morse complex of 