After Carter took five wickets to help wrap up the home first innings for 392, Notts were dismissed for just 134.
Following on, they made a much better effort, ending day two on 112 without loss, still trailing by 246. Jake Libby and Steven Mullanery were both unbeaten on 54.
Carter said: “The last time I took five wickets was here on my debut so I am buzzing to have finally done so again.
“It didn’t take us too long to finish the Somerset innings, but we were very disappointed with the way we batted. To then score 112 without losing a wicket was a great effort.
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“The pitch is turning increasingly so if we can get a lead of between 150 and 250 it will give us a chance of getting over the line. I am also buzzing for Jake Libby, having seen the work he has put in without getting the scores he wanted. Hopefully, he can carry on and get a big one tomorrow.”
Craig Overton gave the England selectors a reminder of his talent with some impressive pace bowling, claiming for 53 to send the visitors crumbling to 134 — Ross Taylor top-scoring with 74.
The day began with Somerset 307 for seven, with 13 overs in which to secure a fourth batting point. They fell nine runs short, while Notts could only take on wicket in that time, so the bonus points were three to two in favour of the hosts.
Steve Davies, unbeaten on 59 overnight, received good support from Dom Bess (34) before the latter edged Samit Patel to slip with the score on 340.
Davies again looked rock solid, but Overton played a poor shot to present Matt Carter (five for 113) with his fifth wicket and when Josh Davey fell lbw to Patel for 14, Davies was left unbeaten on 92.
Somerset had added 85 to their total and either side of lunch Overton produced an inspired spell from the River End, which saw him take three wickets for four runs in his first three overs.
The tall seamer sent back Libby, Mullaney and Samit Patel, while Lewis Gregory accounted for Chris Nash and Riki Wessels as Notts crashed to 28 for five.
It could have been been worse for the visitors as at that point Abell missed the stumps with a chance to run out Tom Moores. Soon afterwards Taylor, on 14, survived a huge lbw shout from Gregory that looked close.
Abell has been proving a handy extra seamer for Somerset this season and the captain again chipped in with the wickets of Moores and Billy Root, who became the fifth Notts batsman to register a duck.
Taylor needed some fortune to reach 70-ball half-century, but was last man out, having hit Overton for three sixes in an over, one of them clearing the short leg-side boundary comfortably enough to hit the roof of the flats behind the Somerset Stand.
Overton finally ended Taylor’s resistance, reward for running in aggressively and bowling with excellent rhythm and pace throughout his 12.1 overs.
The ball had swung for the Somerset seamers, something that failed to happen in the second innings as Mullaney and Libby launched the fightback in impressive style.
Libby went to 50 off 63 balls, with nine fours, and Mullaney followed to the same landmark off 84 deliveries with ten boundaries.
There was evidence of turn for Roelof van der Merwe, who beat the bat on several occasions, bowling from the River End.