It is known that the moduli space of smooth Fano-Mukai fourfolds $V_{18}$ of genus $10$ has dimension one. We show that any such fourfold is a completion of $\CC^4$ in two different ways. Up to isomorphism, there is a unique fourfold $V_{18}^{\s}$ acted upon by $\SL_2(\CC)$. The group $\Aut(V_{18}^{\s})$ is an extension of $\GL_2(\CC)$ by $\ZZ/2\ZZ$. Furthermore, $V_{18}^{\s}$ is a $\GL_2(\CC)$-equivariant completion of $\CC^4$, and as well of $\GL_2(\CC)$. The restriction of the $\GL_2(\CC)$-action on $V_{18}^{\s}$ to $\CC^4\hookrightarrow V_{18}^{\s}$ yields a faithful representation with an open orbit. There is also a unique, up to isomorphism, fourfold $V_{18}^{\aaa}$ such that the group $\Aut(V_{18}^{\aaa})$ is an extension of $\Ga\times\Gm$ by $\ZZ/2\ZZ$. For a Fano-Mukai fourfold $V_{18}$ neither isomorphic to $V_{18}^{\s}$, nor to $V_{18}^{\aaa}$, one has $\Aut^0 (V_{18})\cong (\Gm)^2$, and $\Aut(V_{18})/\Aut^0(V_{18})$ is a cyclic group whose order is a factor of $6$.