It's simply because 許される is passive.
真那に起動を許す
真那に起動が許される
cf.
彼女に手紙を渡す
彼女に手紙が渡される
彼に事情を話す
彼に事情が話される
EDIT:
Additional information: 真那に起動を許される is also valid as a passive sentence. In this case, 真那 is the one who allows to activate/boot it, and the speaker or someone else is the one who is allowed to do it.