The correct answer is C. for.
The word “for” is used to indicate a purpose or goal. In this case, the speaker is saying that the woman is very keen on going abroad for higher studies. This means that she has a strong desire to go abroad in order to further her education.
The other options are not as appropriate in this context. The word “at” is used to indicate a location or position. The word “on” is used to indicate a surface or area. The word “over” is used to indicate a period of time or a distance.
In conclusion, the correct answer is C. for. This is because the word “for” is used to indicate a purpose or goal. In this case, the speaker is saying that the woman is very keen on going abroad for higher studies.