  Let $K$ be a finite, connected, abstract simplicial complex. The Morse
complex of $K$, first introduced by Chari and Joswig, is the simplicial complex
constructed from all gradient vector fields on $K$. We show that if $K$ is
neither the boundary of the $n$-simplex nor a cycle, then
$\mathrm{Aut}(\mathcal{M}(K))\cong \mathrm{Aut}(K)$. In the case where $K=
C_n$, a cycle of length $n$, we show that $\mathrm{Aut}(\mathcal{M}(C_n))\cong
\mathrm{Aut}(C_{2n})$. In the case where $K=\partial\Delta^n$, we prove that
$\mathrm{Aut}(\mathcal{M}(\partial\Delta^n))\cong
\mathrm{Aut}(\partial\Delta^n)\times \mathbb{Z}_2$. These results are based on
recent work of Capitelli and Minian.
