is Proposition \[prop: aut(M(K))=aut(K)\] where we prove that ${\mathrm{Aut}}({\mathcal{M}}(K))\cong {\mathrm{Aut}}(K)$ for $K\neq C_n, {\partial\Delta}^n.$ In Section \[Induced Maps\], we show that one can induce an automorphism on ${\mathcal{M}}(K)$ from an automorphism of $K$ . We then show that there is an injection of ${\mathrm{Aut}}(K)$ into ${\mathrm{Aut}}({\mathcal{M}}(K))$ in Proposition \[prop: autK subgroup\]. We are then able to show that ${\mathrm{Aut}}({\mathcal{M}}(K))\cong {\mathrm{Aut}}(K)$ for $K\neq {\partial\Delta}^n, C_n$ by utilizing results of Capitelli and Minian [@CM-17]. These results concern when we may pull an automorphism of the Morse complex back to an automorphism of the original complex. The case where $K=C_n$ follows as a corollary of the more general fact that ${\mathrm{Aut}}({\mathcal{M}}(K))\cong {\mathrm{Aut}}({\mathcal{H}}(K))$ where ${\mathcal{H}}(K)$ is the Hasse diagram of $K$. We establish this later isomorphism in Proposition \[prop: aut(M(K))=aut(H(K))\] and prove that ${\mathrm{Aut}}({\mathcal{M}}(C_n))\cong {\mathrm{Aut}}(C_{2n})$ in Proposition \[prop: aut C\_n\].

The case where $K={\partial\Delta}^n$ is Proposition \[prop: aut(M(K)) pt2\]. In this case, there are automorphisms of the Morse complex which are not induced by a simplicial map on the original complex, called ghost automorphisms. We define what we call the reflection map $\pi$ which is a cosimplicial map in the sense that if $\alpha\subseteq \beta$, then $\pi(\alpha) \supseteq \pi(\beta)$. This cosimplicial map induces and 