The solutions of Einstein's equation that you can find in GR books all deal with highly idealized situations. Examples: The FLRW solutions are what you get when you assume that spacetime can be sliced into a one-parameter family of spacelike hypersurfaces that are all homogeneous and isotropic. (We can think of as them as representing space at different times). There are three such solutions. All of them describe an expanding universe. The Schwarzschild solution is what you get when you assume that spacetime is completely empty except for a spherical, non-rotating distribution of mass that has existed forever, and will continue to exist forever. There is only one such solution. The universe it describes isn't expanding.
If we're trying to describe spacetime near a star, it's pointless to use anything but the Schwarzschild solution. Spacetime can't be exactly Schwarzschild of course, since there are other things in the universe, but it's going to be a lot more like Schwarzshild than like FLRW for example, since space isn't at all homogeneous near the star. The universe is only homogeneous on very large scales.
So the correct solution describing spacetime near a star of finite life span in a universe where there are lots of other stars, distributed in galaxies and clusters of galaxies such that the large-scale structure is approximately homogeneous and isotropic, should almost certainly contain some amount of expansion, but it's likely to be ridiculously tiny. (Actually the expansion per year of a region the size of the solar system is very small in the FLRW solutions too. What I meant is that the expansion in the correct solution is likely to be ridiculously tiny compared to that).
I'm saying "likely" because I don't think these things have been proved conclusively yet.
