  Let $G$ be a finite abelian group. The critical number ${\rm cr}(G)$ of $G$
is the least positive integer $\ell$ such that every subset $A\subseteq
G\setminus\{0\}$ of cardinality at least $\ell$ spans $G$, i.e., every element
of $G$ can be written as a nonempty sum of distinct elements of $A$. The exact
values of the critical number have been completely determined recently for all
finite abelian groups. The structure of these sets of cardinality ${\rm
cr}(G)-1$ which fail to span $G$ has also been characterized except for the
case that $|G|$ is an even number and the case that $|G|=pq$ with $p,q$ are
primes. In this paper, we characterize these extremal subsets for $|G|\geq 36$
is an even number, or $|G|=pq$ with $p,q$ are primes and $q\geq 2p+3$.
