  We consider deformations of quantum exterior algebras extended by finite
groups. Among these deformations are a class of algebras which we call
truncated quantum Drinfeld Hecke algebras in view of their relation to
classical Drinfeld Hecke algebras. We give the necessary and sufficient
conditions for which these algebras occur, using Bergman's Diamond Lemma. We
compute the relevant Hochschild cohomology to make explicit the connection
between Hochschild cohomology and truncated quantum Drinfeld Hecke algebras. To
demonstrate the variance of the allowed algebras, we compute both classical
type examples and demonstrate an example that does not arise as a factor
algebra of a quantum Drinfeld Hecke algebra.
