Ask for = ask to speak to somebody; ask for news of/about something, ask for something.

Ask somebody in/out =invite somebody in (at home) or out (to a restaurant, to the cinema, etc..).

Prasal verb Translation Examples
Ask for somebody Chiedere di qualcuno Go to the library and ask for the director.
Ask after Chiedere di qualcuno I met Elizabeth at the theatre and she asked after you.
Ask somebody in Invitare qualcuno After the match I asked all my friends in.
Ask somebody out Invitare qualcuno fuori  I phoned her and asked her out to dinner.