The word averiguaban is the imperfect form of averiguar in the third person plural.
See the full averiguar conjugation

averiguar

Me dijeron que me llamarían si averiguaban algo.
They told me they would get back to me if anything turned up.
Si lo averiguaban, lo perdería todo, todo lo que he conseguido como Thomas Pope.
If they found out, I would lose everything, everything that I have built as Thomas Pope.
Estando aún con vida algunos de los que averiguaban acerca del futuro, ordenaría milagrosamente los asuntos de las naciones dominantes en la tierra, y daría ascendencia a los babilonios.
Within the lifetime of some who were even then making inquiry regarding the future, He would miraculously shape the affairs of the ruling nations of earth and bring the Babylonians into the ascendancy.
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