The Polish space Y constructed in [vM1] admits no nontrivial isotopy. Yet, there exists a Polish group that acts transitively on Y.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-1-8, author = {Tadeusz Dobrowolski}, title = {A Polish AR-Space with no Nontrivial Isotopy}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {56}, year = {2008}, pages = {67-74}, zbl = {1148.57304}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-1-8} }
Tadeusz Dobrowolski. A Polish AR-Space with no Nontrivial Isotopy. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) pp. 67-74. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-1-8/