A Constitutional guide to the issue:
According to CRS, the inherent contempt power is uniquely characterized by "not requiring the cooperation or assistance of either the executive or judicial branches."
While it is true that the President can immunize persons from criminal prosecution, it does not appear that he has authority to immunize a witness from congressional inherent contempt proceeding. . . . A finding of inherent contempt against an executive branch official does not appear to be subject to the President’s Pardon power ...
Although the contemnor can seek judicial review by means of a petition for a writ of habeas corpus, the scope of such review may be relatively limited, compared to the plenary review accorded by the courts in cases of conviction under the criminal contempt statute.