The last 3 years each summer the cat got fleas even though she had frontline treatment. She would bring them in and they would get into the rugs furniture before the frontline got to them, then they looked for fresh meat. I'm not sure what was carrying them outside? Fleas can be bad as they are plague vectors.Ticks are really bad here right now as are fleas and I haven’t seen fleas like this in years! Be careful out there.
The permethrin spray will also work on carpets/couches for fleas. In fact they sell it for that purpose but the diluted stuff is ~ $20 gallon. TSC carries the carpet/couch spray.
Again it is not cat friendly so let things dry before you let kitty back.
One more thing on permethrin, it kills most insects, bad and good. That's why I won't use it as a broadcast spray in my yard. I have a lot of beneficial (ladybugs, parasitic wasps, dragonflies, bees, praying mantis, etc....) that I want to help me combat the nasties