For each countable residually finite group G, we present examples of
irregular Toeplitz subshifts in {0,1}G that are topo-isomorphic extensions
of its maximal equicontinuous factor. To achieve this, we first establish
sufficient conditions for Toeplitz subshifts to have invariant probability
measures as limit points of periodic invariant measures of {0,1}G. Next,
we demonstrate that the set of Toeplitz subshifts satisfying these conditions
is non-empty. When the acting group G is amenable, this construction provides
non-regular extensions of totally disconnected metric compactifications of G
that are (Weyl) mean-quicontinuous dynamical systems.Comment: 23 pages, 4 figure