Learn Farsi: گروه هم‌زمان شروع کردند.

This is exactly how collective nouns for people work in Farsi! Words like گروه (group) or خانواده (family) take plural verbs when referring to the actions of the people inside them. Even though گروه is structurally singular, we use the 3rd-person plural verb کردند because the individuals in the group are doing the starting.

Could I say گروه‌ها to make the subject match the plural verb?

You could, but it would change the meaning to "The groups started simultaneously" (referring to multiple different groups). To talk about a single group of people, you must keep the noun singular (گروه) while using the plural verb (شروع کردند).

How is the word هم‌زمان constructed?

It combines the prefix هم (same / together) and the noun زمان (time). Literally meaning "same-time", it works perfectly here as an adverb for "simultaneously". You'll see هم used a lot to show sharing or togetherness, like in هم‌کلاس (classmate).

Why does هم‌زمان come before شروع instead of at the end of the sentence like in English?

Farsi has a Subject-Object-Verb word order, meaning verbs almost always go at the very end. Adverbs of manner or time like هم‌زمان usually sit right before the verb. Since شروع کردند is a compound verb (made of a noun and a "helper" verb), the adverb goes right before that entire block.