  Taking into account the Schuster-Toro action and its fermionic analogue
discovered by us, we supersymmetrize unconstrained formulation of the
continuous spin gauge field theory. Afterwards, building on the Metsaev
actions, we supersymmetrize constrained formulation of the theory. In each
formulation, we provide supersymmetry transformations for the $\mathcal{N}=1$
supermultiplet in four-dimensional flat space-time, in which continuous spin
particle (CSP) is considered to be a complex scalar continuous spin field, and
its superpartner which can be called ``\,CSPino\,'' is considered to be a Dirac
continuous spin field. It is shown that the algebra of these supersymmetry
transformations are closed on-shell. Furthermore, we investigate whether
obtained supersymmetry transformations reproduce the known result of the higher
spin gauge field theory in the helicity limit. Finally, we illustrate how these
two separate set of obtained transformations are related to each other.
