al. have shown that the pure Morse complex of a tree is collapsible and some other results on the pure Morse complex of an arbitrary graph [@A-F-Q-V-08]. Kozlov studied shellability and other properties for trees [@Kozlov99], although the language of the Morse complex was not available to him at the time. Recently, Capitelli and Minian showed that the isomorphism type of the Morse complex completely determines the isomorphism type of the corresponding simplicial complex [@CM-17]. Other than these results, very little is known about the Morse complex. Its sheer size alone makes it a notoriously complex object of study.

The goal of this paper is to compute the automorphism group of the Morse complex ${\mathcal{M}}(K)$. We derive a formula relating ${\mathrm{Aut}}({\mathcal{M}}(K))$ to ${\mathrm{Aut}}(K)$ for $K$ any finite, connected, abstract simplicial complex. Our main result is the following:

\[thm: main\] Let $K$ be a finite, connected, abstract simplicial complex. Then $${\mathrm{Aut}}({\mathcal{M}}(K))\cong
\begin{cases}
{\mathrm{Aut}}(K) & \text{if } K\neq {\partial\Delta}^n,C_n \\
{\mathrm{Aut}}(C_{2n}) & \text{if } K= C_n \\
{\mathrm{Aut}}(K)\times \mathbb{Z}_2 & \text{if } K= {\partial\Delta}^n. \\
\end{cases}$$

Here ${\partial\Delta}^n$ is the boundary of the $n$-simplex and $C_n$ is the cycle of length $n$. Theorem \[thm: main\] is proved in three parts. The first 