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We study the inclusion of fermionic degrees of freedom and ${\cal N}=1$supersymmetry in 4-dimensional manifolds with arbitrary torsion andnon-metricity tensors deforming the connection. We inspect the closure of localdiffeomorphism, Lorentz and supersymmetric transformations on the vielbein andgravitino at leading order in fermions. The closure of a pair of supersymmetrictransformations acting on the vielbein restricts the form of the spinconnection. The dependency of the latter on the fundamental fields isexclusively given by the vielbein, up to Lorentz and/or diffeomorphismsbrackets, independently of the torsion or non-metricity deformations of theaffine connection, which is not restricted by supersymmetry at this order. Wehave then studied the couple of the Rarita-Schwinger theory to a genericgravitational Lagrangian; the supersymmetric variation of the former restrictsthe form of the bosonic Lagrangian in order to accomplish invariance. Specialcases where $R_{\mu \nu}{}^{ab}=0$, $Q_{\mu \nu \rho}=0$ or $T_{\mu\nu}{}^{\rho}=0$ are provided.

${\cal N}=1$ supersymmetry in $4D$ gravity with torsion and non-metricity