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 ${\mathrm{\Lambda }}_{\wedge }[H]$ (the completion of the localization at $l$ of ${\mathbb{Z}}_l[[T]][H]$) as well as the kernel and cokernel of the integral logarithm $L:{\mathrm{\Lambda }}_{\wedge }[H{]}^{\times }\to {\mathrm{\Lambda }}_{\wedge }[H]$, which appears in non-commutative Iwasawa theory.