The Whitehead group of (almost) extra-special p-groups with p odd

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P , the subgroup Cl_1 (ZP) of SK_1 (ZP), in terms of a genetic basis of P. We also introduce a deflation map Cl_1 (ZP) $\to$ Cl_1 (Z(P/N)) , for a normal subgroup N of P , and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of SK_1 (ZP), when P is an elementary abelian p-group.