This document "How is inflation measured?" from the Riksbank: https://www.riksbank.se/globalassets/media/rapporter/ekonomi... doesn't give weights in general but does say this (emphasis mine): "the method in which housing costs for owner occupied housing are measured are an important source of non-comparability, partly because they make up a large proportion of the CPI (around 10 per cent in sweden)". Note the "owner occupied" part, so this isn't even just rents! But then I don't understand the rest of the discussion that follows this.
Also Statistics Sweden at https://www.scb.se/en/finding-statistics/statistics-by-subje... lists "Housing, water, electricity, and fuels" as one of the "main groups" in the CPI, though this only suggests but does not prove that housing does have a non-zero weight.
When taking about inflation it's also always good to remind one self that the pool of money also changes. And the fact that housing usually is connected to mortgages, which in turn expands the amount of available money. But this paragraph has nothing to do with your comment so I'll just stop there!