  We compute the abelianization of the Jennings group
$\mathcal{J}_k(\mathbb{Z})$ of powers series with constant coefficient $0$,
linear coefficent equal to $1$ and vanishing coefficients in orders greater or
equal than $2$ and less than $k$, where $k\geqslant2$. This is accomplished by
directly dealing with the equivalence classes in the corresponding
abelianizations, in contrast with the work of I. K. Babenko and S. A. Bogatyy,
who give an explicit abelianization morphism for the case $k=2$.
