  Extending ideas of twisted equivariant $K$-theory, we construct twisted
versions of the representation rings for Lie superalgebras and Lie supergroups,
built from projective $\Z_{2}$-graded representations with a given cocycle. We
then investigate the pullback and pushforward maps on these representation
rings (and their completions) associated to homomorphisms of Lie superalgebras
and Lie supergroups.
  As an application, we consider the Lie supergroup $\Pi (T^{*}G)$, obtained by
taking the cotangent bundle of a compact Lie group and reversing the parity of
its fibers. An inclusion $H \hookrightarrow G$ induces a homomorphism from the
twisted representation ring of $\Pi(T^{*}H)$ to the twisted representation ring
of $\Pi(T^{*}G)$, which pulls back via an algebraic version of the Thom
isomorphism to give an additive homomorphism from $K_{H}(\mathrm{pt})$ to
$K_{G}(\mathrm{pt})$ (possibly with twistings). We then show that this
homomorphism is in fact Dirac induction, which takes an $H$-module $U$ to the
$G$-equivariant index of the Dirac operator $\dirac \otimes U$ on the
homogeneous space $G/H$ with values in the homogeneous bundle induced by $U$.
