noun
two sets; two classes; double
Refers to two groups, pairs, or sets of things; can also mean 'double' in contexts like double suicide (心中). The gloss 'double' is less common as a standalone meaning.
トランプを二組買ってきた。
I bought two packs of playing cards.
この学校には一年生が二組ある。
This school has two first-year classes.
二つ is the general counter for two items, while 二組 specifically refers to two sets, pairs, or groups.
二倍 means 'double' in a multiplicative sense (twice as much), whereas 二組 can mean 'double' in the sense of a pair or duo.
Compound of 二 (ふた, 'two') and 組 (くみ, 'set, group, class'). The reading ふたくみ is a regular combination of the native Japanese numerals and the noun.