The integral logarithm in Iwasawa theory: an exercise

Let $l$ be an odd prime number and $H$ a finite abelian $l$-group. We determine the unit group of $\Lambda_\wedge[H]$ (the completion of the localization at $l$ of $\Bbb{Z}_l[[T]][H]$) as well as the kernel and cokernel of the integral logarithm $L:\Lambda_\wedge[H]^\times\to \Lambda_\wedge[H]$, which appears in non-commutative Iwasawa theory.