I am looking for an example of a topological group with countable tightness with the property then it is not metrizable, but every countable subset is metrizable but I cannot construct an example.

This question is inspired by my earlier question
when-is-the-topology-generated-by-countable-subsets
where someone gave me an example of a topological **space** with all of the above properties, namely the 1-point-compactification of an uncountable discrete set and now I was wondering if there are examples of topological spaces with these properties that carry a group strucuture.

Thanks in advance, Tom