  We combine results about Whitehead groups of finite groups with results about
genetic bases of finite $p$-groups to compute the Whitehead groups of some
metacyclic $p$-groups. Let $C_{p^n}$ denote a cyclic group of order $p^n$ for
$p$ an odd prime and $n$ a positive integer. We present a conjecture for the
torsion part of the Whitehead group of $C_{p^n}\times C_{p^n}$ and we show that
torsion part of the Whitehead group of the semidirect product of $C_{p^{n-1}}$
by $C_p$ is isomorphic to $C_p^{(n-2)(p-1)}$. The techniques we use can be
applied to any abelian $p$-group and to many other $p$-groups for $p$ odd.
