Can an institue implement a judgement to part of the applicants

share complete facts

They may go for such stay order, but they are bound to state facts before the court, and if the opposite party bring the attention of the court that the institution has acted with bias by implementing the judgment to some and differing with others, then they may not even entertain such stay order and even pass strictures.  A party can not approbate and reprobate the same statement.

If the institute is aggrieved over any part of the judgment then it can prefer an appeal against the aggrieved portion of judgment alone, it is not necessary that the appeal has to be preferred against the entire judgment only.

The  queriest must post bare minimum facts as guidance can never be given on assumptions.

.What was the judgment?

In what way it was implemented by the party ?

In what way the same party wished to go for  stay order ?

The posting is a riddle than a query. Any meaningful suges that the relevant and complete material facts be stated.