= Solution
The <Noether gauging procedure> starts by allowing a constant <symmetry> parameter to depend on position. For an internal <symmetry> written as $\delta\phi^i=\epsilon^AT_A\phi^i$, integration by parts identifies the <Noether current>:
$$
\delta S_0=\int d^4x\,j_A^\mu\partial_\mu\epsilon^A
$$
up to a boundary term. Introduce a <gauge field> $A_\mu^A$ and add $-gA_\mu^Aj_A^\mu$, with the inhomogeneous <gauge transformation> $\delta A_\mu^A=g^{-1}\partial_\mu\epsilon^A+\cdots$. This cancels the displayed variation. The current coupling has its own variation, so one iterates: add the needed higher-order interactions and corrections to the <gauge transformations>, and require closure of the local <symmetry> algebra. For an ordinary <gauge theory>, this reorganizes derivatives into <gauge covariant derivatives> and fixes the nonlinear <gauge field strength>. Gauging is therefore more than replacing a constant parameter by a function.
For rigid <supersymmetry>, the corresponding <Noether current> is the spinor-valued <supercurrent> $S^\mu$. A local <Majorana spinor> parameter gives a variation proportional to $(\partial_\mu\bar\epsilon)S^\mu$. A <gravitino> $\Psi_\mu$, with an inhomogeneous transformation $\delta\Psi_\mu\sim\kappa^{-1}\partial_\mu\epsilon$, cancels it through a coupling proportional to $\kappa\bar\Psi_\mu S^\mu$. The commutator of two rigid <supersymmetry> transformations is a translation. With local parameters, the translation parameter becomes position dependent, so the completion must also gauge translations: it introduces the <vierbein>, <diffeomorphism invariance of general relativity>, and <local Lorentz transformations>. The <gravitino> and <graviton> thus belong to the same <supergravity multiplet>.
For pure <minimal four-dimensional supergravity>, start with the free <massless Fierz-Pauli action> and the free <Rarita-Schwinger field> kinetic term. The <Noether gauging procedure> replaces the former by the <Einstein-Hilbert action> and couples the latter to the <vierbein> and <spin connection>. In a canonical normalization the leading transformations can be written
$$
\delta e_\mu{}^a=\frac{\kappa}{2}\bar\epsilon\gamma^a\Psi_\mu,
\qquad
\delta\Psi_\mu=\kappa^{-1}\widetilde D_\mu\epsilon.
$$
An overall sign or phase changes with the curvature and <Majorana spinor> conventions, but the relative normalization must be used consistently. To see why these terms fit together, first work to quadratic order in <fermions>, so the <spin connection> is torsion free. Varying the <Rarita-Schwinger field> kinetic term under $\delta\Psi_\mu=\nabla_\mu\epsilon/\kappa$ produces an antisymmetric double <spinor covariant derivative>. The <spinor curvature identity> and the <Einstein tensor contraction of spin curvature> give, in the matching curvature convention,
$$
[\nabla_\rho,\nabla_\sigma]\epsilon
=\frac14R_{\rho\sigma ab}\gamma^{ab}\epsilon,
\qquad
\gamma^{\mu\rho\sigma}\nabla_\rho\nabla_\sigma\epsilon
=\frac12G^\mu{}_{\nu}\gamma^\nu\epsilon.
$$
The resulting <Einstein tensor> term cancels the variation of the <Einstein-Hilbert action> under $\delta e_\mu{}^a$. This is the essential first nontrivial cancellation in the construction.
At higher <fermion> order, the variations of the <vierbein> and <spin connection> produce further terms. Treating the <spin connection> as independent makes its <Euler-Lagrange field equation> algebraic. Its solution is a <gravitino-induced torsion> connection. With $\Psi_a=e_a{}^\mu\Psi_\mu$ and $\gamma^{ab}=[\gamma^a,\gamma^b]/2$, a conventional canonical normalization gives
$$
\widetilde\omega_{\mu ab}=\omega_{\mu ab}(e)+K_{\mu ab},
\qquad
K_{\mu ab}=\frac{\kappa^2}{4}
\left(\bar\Psi_\mu\gamma_a\Psi_b-\bar\Psi_\mu\gamma_b\Psi_a
+\bar\Psi_a\gamma_\mu\Psi_b\right).
$$
Equivalently, with $T^a=de^a+\widetilde\omega^a{}_b\wedge e^b$, the <torsion tensor> obeys
$$
T_{\mu\nu}{}^a=\frac{\kappa^2}{2}\bar\Psi_\mu\gamma^a\Psi_\nu.
$$
The <Majorana Grassmann bilinear interchange> makes the right-hand side antisymmetric in $\mu,\nu$. Eliminating the <spin connection> generates the four-<fermion> completion. In the <1.5-order formalism> one substitutes its solution but varies the other fields holding the connection fixed: the omitted chain-rule term is $(\delta S/\delta\omega)\delta\widetilde\omega=0$. The remaining cubic <fermion> variations cancel by the <Clifford algebra> and <Fierz rearrangements>. This produces the displayed <supergravity> action in its compact connection-dependent form. Calling this formulation <on shell> means that no independent <auxiliary fields> have been retained and that closure of the <supersymmetry> algebra uses the propagating <Euler-Lagrange field equations>; it does not mean that action invariance is obtained only by setting every variation to zero on a solution.
In the antisymmetrized kinetic term, $\Psi_\sigma$ is a spinor-valued one-form and the derivative is conventionally written
$$
\widetilde D_\rho\Psi_\sigma
=\partial_\rho\Psi_\sigma+\frac14\widetilde\omega_{\rho ab}\gamma^{ab}\Psi_\sigma,
\qquad \gamma_\nu=e_\nu{}^a\gamma_a.
$$
This is the Lorentz-covariant exterior-derivative convention for that term. If one instead uses the full affine <covariant derivative> of a vector-spinor, a coordinate-index connection is also present and the antisymmetrization must include its <torsion tensor> correction. The crucial physical distinction from pure metric <general relativity> is \b[the spin connection contains gravitino-induced contorsion, rather than being the torsion-free Levi-Civita connection]. It reduces to the ordinary <spinor covariant derivative> when the <gravitino> vanishes.
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