  In \cite[Problem 72]{Fuchs60} Fuchs asked the following question: which
groups can be the group of units of a commutative ring? In the following years,
some partial answers have been given to this question in particular cases. The
aim of the present paper is to address Fuchs' question when $A$ is a {\it
finite characteristic ring}. The result is a pretty good description of the
groups which can occur as group of units in this case, equipped with examples
showing that there are obstacles to a "short" complete classification. As a
byproduct, we are able to classify all possible cardinalities of the group of
units of a finite characteristic ring, so to answer Ditor's question
\cite{ditor}.
