Institute of CA is clearly a Public Authority as defined in the RTI Act.
Further, Sec 2(f) defines information as:
f) "information" means any material in any form, including records, documents, memos, e-mails, opinions, advices, press releases, circulars, orders, logbooks, contracts, reports, papers, samples, models, data material held in any electronic form and information relating to any private body which can be accessed by a public authority under any other law for the time being in force;
As per your description of the private coaching classes, they do have a relationship with ICA.
Therefore ICA, as a PA, has "access" to information held by the coaching classes.
You can apply for the information to ICA.
› Find content similar to: Are Private Colleges covered under RTI