You can definitely say that the cost of living has risen at different rates for different subpopulations, but you can't say they have experienced different rate

You can definitely say that the cost of living has risen at different rates for different subpopulations, but you can't say they have experienced different rates of inflation. Inflation is change in the general price level of the economy, and an economy is something that we can study at a macroeconomic level. In this sense, unless you regard a social group as being an economy on its own, with its own GDP, balance of payments, and whole set of macroeconomic variables, we can't speak of it as experiencing a particular inflation rate.