The primary motive of the group is revenge for being misinterpreted and subsequently disgraced to the Japanese public by the media.
A-21 was originally formed by Japanese global mercenary Seiji Takechi, who retired from service to start a camp on a remote island specifally for the recovery of lost youths, notably those who had committed serious crimes such as murder. By teaching them survival techniques he learnt as a soldier, Takechi instilled discipline within his charges and gave them a means to escape their troubled pasts and lives of crime.
After media got wind of the island, they performed an unsolicited investigation and tampered with military equipment, creating an accident in which several people were killed. This caused them to create a widespread belief that the purpose of the A-21 was heinous, and tried to demonstrate it by releasing the criminal information of its inhabitants.
"We gained a fortune called confidence. We were so happy when we realized we didn't need to commit crimes any more. But other people didn't think so. One time, a TV station found out about our training. They landed on the island without permission. They touched the equipment and there was an accident. It was such a mess after that. They all accused us no matter what caused it. They called A-21 a terrorist group and had the training facility shut down. They revealed the past of the students. Mine too.
After the group was exposed, Takechi - disgraced and ashamed of what happened to his charges - committed suicide. This further incited his former charges to rebel against Japan and get revenge for their loss.
Seeing an opportunity, Amalgam leader Mr. Gold, who was based in Japan, approached A-21 and gave them the means to attack Tokyo. His motivation was to create and control a number of terrorist activities to justify defence expenditure and keep Japan prosperous, as he had control over several Japanese Security departments and Diet members. This led to A-21 acquiring arm slaves, notably the first Plan-1051 Behemoth, and training up a Lambda Driver capable pilot with the inducement drug TI-971.
After the pilot Takuma Kugayama failed to take a dosage of the drug, his erratic behaviour while en-route to the Behemoth led to his capture by Japanese authorities. He was moved to a special military facility, with Mithril officers from the TDD-1 sent to investigate. Captain Teletha Testarossa, Corporal Yang (in the novel) and Andrey Kalinin went to observe the captive. Unbeknown to them, an A-21 team armed with an Rk-92 Savage piloted by Seina was just about to attack the base to free Takuma. When the attack came and destroyed the facility, Tessa managed to escape with Takuma to Tokyo, leaving a badly wounded Kalinin behind, who was then taken hostage by A-21 and taken back to their base in a ship called the George Clinton.
Seina spoke with Kalinin several times, and found him similar to the late Takechi in his outlook. Kalinin did attempt leverage this to sway A-21 away from making an attack, unsuccessfully.
A-21 meanwhile was able to track and locate Takuma, recapturing him alongside a Mithril team led by Sousuke Sagara. After finally getting Takuma to the Behemoth and activating it, the ship in which it was housed sank, killing or injuring the remaining A-21 members, including Seina, who realised that the groups revenge was about to be inflicted.
We want to paint this town full of peace in our colours. If you call that revenge, you're right. Destroy it completely and cover it with horrifying flames.
After easily destroying a police and military effort to halt the giant AS with Type-96 units, as well as Melissa Mao's M9 sent to monitor events, the Behemoth was engaged by Sagara in the ARX-7 Arbalest and destroyed, along with its pilot, effectively ending A-21.