Group first to object about tweet oddly (5)
I believe the answer is:
octet
'group' is the definition.
(octet is a kind of group)
'first to object about tweet oddly' is the wordplay.
'first to' says to take the initial letters.
'about' becomes 'c' (abbreviation for 'circa'**).
'oddly' means one should take alternating letters.
The initial letter of 'object' is 'o'.
The alternate letters of 'tweet' are 'tet'.
'o'+'c'+'tet'='OCTET'
(Other definitions for octet that I've seen before include "music group" , "A group of eight instruments or voices" , "Group of eight players" , "Musical composition for eight performers" , "A number of musicians" .)