The problem of determining whether several finite automata accept a word in common is closely related to the well-studied membership problem in transformation monoids. We raise the issue of limiting the number of final states in the automata intersection problem. For automata with two final states, we show the problem to be ?L-complete or NP-complete according to whether a nontrivial monoid other than a direct product of cyclic groups of order 2 is allowed in the automata.We further consider idempotent commutative automata and (abelian, mainly) group automata with one, two or three final states over a singleton or larger alphabet, elucidating the complexity of the intersection nonemptiness and related problems in each case.