Single-symbol insertion into a regular language ($\{xay:xy\in L\}$)

For a fixed symbol $a$ and a regular language $L$, inserting exactly one $a$ at an arbitrary position preserves regularity. Make two copies of a recognizing finite-state automaton, one before and one after the insertion. Preserve its ordinary automaton transitions in each copy, and add a transition consuming $a$ from each before-state to its matching after-state. Accept only in the after-copy. Every accepting run makes exactly one of the additional moves.