The cleanest way to answer this is to compare what a District Court can do with what only the High Court can do under the Code's transfer scheme. A District Court's transfer power under Section 24 is confined to suits pending in courts subordinate to it, all of which sit within its own district. Once the transfer crosses into another district, that power runs out and only the High Court can step in.
Comparing the District Court's own-district reach against the High Court's State-wide reach shows that only the High Court can order a transfer between courts of two different districts.
Thinking about where such a transfer application is actually filed, and who has the administrative authority to act on it, is a practical way to reach the answer.
Tracing where such an application would practically be filed and decided confirms that inter-district transfer within a State is a matter for the High Court.
The correct answer is therefore By the High Court.