How do you eliminate any possible ambiguity from the following sentence?

Within each group, everybody spoke the same language.

Language use was homogeneous within each group

Each group had a common language, spoken by all members. No ambiguity, no fancy words. Conveys the message clearly and very less chance of it being misunderstood.

How do you know that what is meant is that the language is common within groups and not between them? Maybe it would be clearer if you said “spoken by all its members”.

Use the KISS theory “Each group spoke the same language.”

Does that mean that every group spoke the same language as every other group?

If the groups all spoke the same language you could say “everyone spoke the same language.”

If each group spoke a different language then “each group spoke a different language but within each group everyone spoke the same language”.

If only some groups spoke a different language you could say “some groups spoke different languages but within each group the language was the same.”

Everyone in each group shared a common language with the other members of that group.

I suppose @Jeruba‘s is most succinct without more context.

We’ll break into same language small groups.