The fouh dorsal interosseous muscle is innervated by:

The deep branch of the ulnar nerve innervates the intrinsic muscles of the hand (with the exception of the thenar compament). This includes the dorsal interosseous muscles, the palmar interosseous muscles, the two lumbrical muscles on the medial side of the hand, and the muscles to the 5th digit (digiti minimi). The dorsal branch of the ulnar nerve innervates the skin of the dorsal surface of the medial 1.5 digits and the skin of the medial side of the back of the hand. The recurrent motor branch of the median nerve innervates the thenar compament. The superficial branch of the radial nerve provides sensory innervation to the skin on the radial side of the dorsum of the hand, including the radial 3.5 digits. Finally, the superficial branch of the ulnar nerve innervates the skin of the palmar surface of the medial 1.5 digits as well as the skin of the medial side of the front of the hand. Ref: Moon D.A., Foreman K.B., Albeine K.H. (2011). Chapter 33. Hand. In D.A. Moon, K.B. Foreman, K.H. Albeine (Eds), The Big Picture: Gross Anatomy.