KS wrote:

My kitchen has some sort of indoor/outdoor carpet that seems to be very
securely glued to the floor. A test showed that this will take a lot of
muscle to pull up and there will be a substantial layer of glue left
behind. Can I put vinyl tiles over the glue residue, provided that the
residue is fairly uniform? Or will I have to put down 1/4" plywood to
provide a more suitable sub-surface? Any other way to prep the floor for
vinyl flooring?

Any defect in the subfloor will eventually be seen in the vinyl - and that
includes seams in any new subfloor, if the subfloor isn't done correctly...