  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
$\mathbb{C}^4$ in two different ways. Up to isomorphism, there is a unique
fourfold $V_{18}^{\mathrm s}$ acted upon by $\operatorname{SL}_2(\mathbb{C})$.
The group $\operatorname{Aut}(V_{18}^{\mathrm s})$ is a semidirect product
$\operatorname{GL}_2(\mathbb{C})\rtimes(\mathbb{Z}/2\mathbb{Z})$. Furthermore,
$V_{18}^{\mathrm s}$ is a $\operatorname{GL}_2(\mathbb{C})$-equivariant
completion of $\mathbb{C}^4$, and as well of $\operatorname{GL}_2(\mathbb{C})$.
The restriction of the $\operatorname{GL}_2(\mathbb{C})$-action on
$V_{18}^{\mathrm s}$ to $\mathbb{C}^4\hookrightarrow V_{18}^{\mathrm s}$ yields
a faithful representation with an open orbit. There is also a unique, up to
isomorphism, fourfold $V_{18}^{\mathrm a}$ such that the group
$\operatorname{Aut}(V_{18}^{\mathrm a})$ is a semidirect product $({\mathbb
G}_{\mathrm{a}}\times{\mathbb G}_{\mathrm{m}})\rtimes
(\mathbb{Z}/2\mathbb{Z})$. For a Fano-Mukai fourfold $V_{18}$ neither
isomorphic to $V_{18}^{\mathrm s}$, nor to $V_{18}^{\mathrm a}$, one has
$\operatorname{Aut}^0 (V_{18})\cong ({\mathbb G}_{\mathrm{m}})^2$, and
$\operatorname{Aut}(V_{18})$ is a semidirect product of
$\operatorname{Aut}^0(V_{18})$ and a finite cyclic group whose order is a
factor of $6$.
