it must be something to do with the meaning and the conventions governing how these words are used .. but I don't know exactly what the technical terms would be
for the sentence you wrote you have to use denied, it is "the admission" which is being denied ...
You COULD say:"They were rejected because they were foreigners"
Deny means to declare untrue or refuse to accept as a fact. It also means to refuse to give or allow. “She denies any involvement in the murder.” “She was denied the chance of going to university when her parents’ business collapsed.”
Reject means to refuse to accept, consider or use. “She rejected my suggestion.” “If people are unkind to him he feels rejected.”