The purpose of this paper is: (i) to construct a space which is semilocally
simply connected in the sense of Spanier even though its Spanier group is
non-trivial; (ii) to propose a modification of the notion of a Spanier group so
that via the modified Spanier group semilocal simple connectivity can be
characterized; and (iii) to point out that with just a slightly modified
definition of semilocal simple connectivity which is sometimes also used in
literature, the classical Spanier group gives the correct characterization
within the general class of path-connected topological spaces. While the
condition "semilocally simply connected" plays a crucial role in classical
covering theory, in generalized covering theory one needs to consider the
condition "homotopically Hausdorff" instead. The paper also discusses which
implications hold between all of the abovementioned conditions and, via the
modified Spanier groups, it also unveils the weakest so far known algebraic
characterization for the existence of generalized covering spaces as introduced
by Fischer and Zastrow. For most of the implications, the paper also proves the
non-reversibility by providing the corresponding examples. Some of them rely on
spaces that are newly constructed in this paper.