In this paper a condition for
two central decompositions of a group into centrally indecomposable factors to be
isomorphic modulo the center is derived. Using this result a larger class of groups
with central decompositions into centrally indecomposable factors isomorphic
modulo the center than that of an earlier work of the author is determined.
An example of a group having two central decompositions into centrally
indecomposable factors which are not isomorphic modulo the center is also
obtained.