in this context there has also been a growing interest to extend AdS and Poincaré supergravity theories to other symmetries (see [@Concha:2018jxx; @Concha:2019icz] and references therein).

In this section, we briefly review (following [@Concha:2016zdb]) the construction of the $\mathfrak{osp}(2|1)\otimes \mathfrak{sp}(2)$ CS supergravity (three-dimensional AdS CS supergravity), and we pave the way for the development of the CS supergravity action in three spacetime dimensions invariant under the $D=3$ AdS Carroll superalgebra which will be presented in Section \[S2\].

The three-dimensional CS action is given by $$\label{genCS}
I_{CS}=\frac{k}{4 \pi} \int_\mathcal{M} \Big \langle A dA + \frac{2}{3} A^3 \Big \rangle \, ,$$ where $k=1/(4G)$ is the CS level of the theory (and for gravitational theories is related to the gravitational constant $G$), $A$ corresponds to the gauge connection $1$-form, $\langle \ldots \rangle$ denotes the invariant tensor, and the integral is over a three-dimensional manifold $\mathcal{M}$.

In the case of $\mathfrak{osp}(2|1)\otimes \mathfrak{sp}(2)$, the connection $1$-form reads $$\label{connection}
\tilde{A} =\tilde{A}^A \tilde{T}_A = \frac{1}{2} \tilde{\omega}^{AB} \tilde{J}_{AB} + \tilde{V}^A \tilde{P}_A + \tilde{\psi}^\alpha \tilde{Q}_\alpha  \,  ,$$ where $\tilde{T}_A = \lbrace \tilde{J}_{AB}, \tilde{P}_A , \tilde{Q}_\alpha \rbrace$ (Lorentz generators, spacetime translations, and a $2$-components Majorana spinor charge, respectively) are the $\mathfrak{osp}(2|1)\otimes \mathfrak{sp}(2)$ generators (with $A,B=0,1,2$ and $\alpha=1,2$) and $\tilde{\omega}^{AB}$, $\tilde{V}^A$, 