The short answer is that there is not much to say about the relationship between and a twist where is a cocycle taking values in . (I am going to write for the twist instead of Serre's notation for the sake of easy typesetting.) You can get a good feel for what is possible by soaking in sections I.5.7 and III.1.4 of Serre's Galois Cohomology.
Section I.5.7
One thing you can do -- as exhibited in I.5.7 -- is twist all three terms in a short exact sequence of -modules and get a new short exact sequence, assuming the obvious compatibility conditions hold. Serre starts with an exact sequence
where is assumed central in . Then he fixes a 1-cocycle with values in and twists to get an exact sequence
.
Note that this twist is not an inner twist of , because need not be in the image of .
This may look like a lame example, in that the twist of is "pretty close" to being inner. But already here you don't have any results regarding a connection between and . That's a pretty fuzzy statement; Serre says as much as you can say with precision in Remark 1: "it is, in general, false that is in bijective correspondence with ."
Section III.1.4
This section discusses your question for the specific case where is the absolute Galois group of a field and is the group of -by- matrices of determinant 1 with entries in a separable closure of . Serre explains what you get as when you twist by a cocycle with values in . You can get, for example, a special unitary group.
You can find explicit descriptions of for some 's in The Book of Involutions, pages 393 (Cor. 29.4) and 404 (box in middle of page). Note that for as in Section I.5.7, is a group (a nice coincidence) but in the case where you get a true special unitary group, does not have a reasonable group structure--it is just a pointed set like you expect.
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