  We investigate the Weyl-Wigner-Gr\"oenewold-Moyal, the Stratonovich and the
Berezin group quantization schemes for the space-space noncommutative
Heisenberg-Weyl group. We show that the $\star$-product for the deformed
algebra of Weyl functions for the first scheme is different than that for the
other two, even though their respective quantum mechanics' are equivalent as
far as expectation values are concerned, provided that some additional criteria
are imposed on the implementation of this process. We also show that it is the
$\star$-product associated with the Stratonovich and the Berezin formalisms
that correctly gives the Weyl symbol of a product of operators in terms of the
deformed product of their corresponding Weyl symbols. To conclude, we derive
the stronger $\star$-valued equations for the 3 quantization schemes considered
and discuss the criteria that are also needed for them to exist.
