François gives the correct affirmative answer. For the negative side, the usual method of proving that the negation of the Continuum Hypothesis is consistent with ZFC is to use the method of forcing to add Aleph_{2} many Cohen reals, so that 2^{ω} = ω_{2} in the forcing extension V[G]. In this model V[G], it is also true that 2^{ω1} = ω_{2}. Thus, this model shows that it is not necessarily true that different-sized sets have different sized power sets. The case of symmetric groups is likely more interesting than free groups, because in this model, the symmetric groups S_{ω} and S_{ω1} have the same cardinality ω_{2}. Nevertheless, these two groups are not isomorphic, as explained in this MO question.

The general answer about what can be true for the continuum function κ --> 2^{κ} is exactly provided by Easton's Theorem. This remarkable theorem states that if you have any class function E, defined on the regular cardinals κ, with the properties that

- κ ≤ λ implies E(κ) ≤ E(λ)
- κ < E(κ)
- κ < Cof(E(κ))

then there is a forcing extension V[G] in which 2^{κ} = E(κ) for all regular cardinal κ. In particular, this shows that the sizes of the power sets (on regular cardinals) are restricted to obey only and exactly the hypotheses listed explicitly above. Each of these properties corresponds to a well-known fact about cardinal exponententiation.

Using Easton's theorem, we can build models of set theory where 2^{κ} = κ^{++} for all regular κ. The added power of the Woodin/Foreman result mentioned by François is that they also get this for singular cardinals.

The point now is that there are innumerable examples provided by Easton's theorem that satisfy your hypothesis that the continuum function is one-to-one. If one begins with a model of GCH and selects *any* injective Easton function E, then the resulting model of set theory V[G] will have E as it's continuum function κ --> 2^{κ} = E(κ) for regular κ, and the GCH will continue to hold at singular κ, preserving injectivity. So one is quite free to satisfy your hypothesis while having any kind of crazy failures of GCH.