And as they went out (Exiontōn de autōn). Genitive absolute with present active participle of exeimi, to go out, old verb, in the N.T. only in Act 13:42; Act 17:15; Act 20:7; Act 27:43. As they (Paul and Barnabas) were going out with all the excitement and hubbub created by the sermon.
They besought (parekaloun). Imperfect active, inchoative, began to beseech. The Textus Receptus inserts wrongly ta ethnē (the Gentiles) as if the Jews were opposed to Paul from the first as some doubtless were. But both Jews and Gentiles asked for the repetition of the sermon (lalēthēnai, first aorist passive infinitive object of parekaloun with accusative of general reference).
The next Sabbath (eis to metaxu sabbaton). Late use (Josephus, Plutarch, etc.) of metaxu (meta and xun=sun) in sense of after or next instead of between (sense of meta prevailing). Note use of eis for “on” or “by.”