Answer is: Yes, one can recover $M$ if it is a compact manifold. See J. V. Whittaker: On Isomorphic groups and homeomorphic spaces, Annals of Math 1963.
EDIT Actually, one knows a lot more, see, for example Tomasz Rybicki Journal: Proc. Amer. Math. Soc. 123 (1995), 303-310. MSC: Primary 58D05; Secondary 17B66, 22E65, 57R50 MathSciNet review: 1233982
And references therein...