Two countably compact topological groups: One of size $\aleph_\omega$ and the other of weight $\aleph_\omega$ without non-trivial convergent sequences

E. K. van Douwen asked in 1980 whether the cardinality of a countably compact group must have uncountable cofinality in $\mathrm{ZFC}$. He had shown that this was true under GCH. We answer his question in the negative. V. I. Malykhin and L. B. Shapiro showed in 1985 that under GCH the weight of a pseudocompact group without non-trivial convergent sequences cannot have countable cofinality and showed that there is a forcing model in which there exists a pseudocompact group without non-trivial convergent sequences whose weight is $\omega_1<{\mathfrak c}$. We show that it is consistent that there exists a countably compact group without non-trivial convergent sequences whose weight is $\aleph_\omega$.