  We construct smooth $\mathbb{C}^*$-actions on the moduli spaces of super
$J$-holomorphic curves as well as super stable curves and super stable maps of
genus zero and fixed tree type such that their reduced spaces are torus
invariant. Furthermore, we give explicit descriptions of the normal bundles to
the fixed loci in terms of spinor bundles and their sections. Main steps to the
construction of the $\mathbb{C}^*$-action are the proof that the charts of the
moduli space of super $J$-holomorphic curves obtained by the implicit function
theorem yield a smooth split atlas and a detailed study of the superconformal
automorphism group of $\mathbb{P}_{\mathbb{C}}^{1|1}$ and its action on
component fields.
