One of the reasons why I like Robertson’s big grammar is stuff like this:

(f) THE QUESTION OF Αὑτοῦ.  This is somewhat knotty. It seems clear that as a rule αὐτοῦ and not αὑτοῦ is to be printed in the N. T. A number of reasons converge on this point. The older Greek often used αὑτοῦ rather than ἑαυτοῦ as shown by the aspiration of the prepositions like ἀφʼ αὑτοῦ, etc. In the N. T. there is not a single case of such aspiration after elision save in a few single MSS. Add to this the fact that the N. T. uses the reflexive pronoun much less than the earlier Greek, “with unusual parsimony” (Hort). Besides the personal pronouns of the first and second persons are frequently employed (Buttmann) where the reflexive might have been used. Buttmann urges also the point that in the N. T. we always have σεαυτοῦ, not σαυτοῦ. The earliest uncial MSS. of the N. T. and the LXX that use the diacritical marks belong to the eighth century, but they all have αὐτοῦ, not αὑτοῦ. Even in the early times it was largely a matter of individual taste as to whether the personal or the reflexive pronoun was used. Blass (p. 35) indeed decides absolutely against αὑτοῦ. But the matter is not quite so easy, for the κοινή inscriptions give examples of ὑφʼ αὑτοῦ in first century B.C. and A.D. Mayser also gives a number of papyri examples like καθʼ αὑτοῦ, μεθʼ αὑτοῦ, ὑφʼ αὑτῶν, where the matter is beyond dispute. Hort agrees with Winer in thinking that sometimes αὑτοῦ must be read unless one insists on undue harshness in the Greek idiom. He instances Jo. 2:24, αὐτὸς δὲ Ἰησοῦς οὐκ ἐπίστευσεν αὑτὸν αὐτοῖς, and Lu. 23:12, προϋπῆρχον γὰρ ἐν ἔχθρᾳ ὄντες πρὸς αὑτούς. There are other examples where a different meaning will result from the smooth and the rough breathing as in 1 Jo. 5:10 (αὑτῷ), 18 (αὐτόν, αὐτοῦ), Eph. 1:5 (αὐτόν), 10 (αὐτῷ), Col. 1:20 (αὐτόν), 2:15 (αὐτῷ). W. H. print αὑτοῦ about twenty times. Winer leaves the matter “to the cautious judgment of the editors.”
A.T. Robertson, A Grammar of the Greek New Testament in the Light of Historical Research ( (Logos, 1919; 2006)), 226.

BDF was not clear at all when discussing this (§31(1), according to the index).