Group A cars have existed for a long time (at least 20 years although the term GroupA is more recent) but were eclipsed by the monstrous group B cars until 1986 (previously known as Group 4) when the latter were banned mainly due to Henri Toivonen's premature death in Corsica driving a Lancia S4 , Attilio Bettega's accident in a Lancia 037 (1985) and a dramatic incident involving a Ford RS200 driven by Joaquim Santos and spectators in the 1986 Portuguese rally. GroupB cars at the time had to be produced to only 200 samples in 12 consecutive months to get the FIA homologation and be able to race. The limited production numbers allowed manufacturers to produce cars whose overall cost would be prohibitive if they were to be produced massively. These cars' only purpose in life was rallying.
Group B cars had in common:
The most renowned Group B representatives were (a full list can be found here):
The cars sometimes had 600+ Bhp engines and a weight below the ton mark. One can easily comprehend the danger the drivers and the spectators were facing. The FIA decided to ban GroupB cars in rallying starting in 1986. Some evolutions of old Group B cars are still being used in rallycross events and others in the European Mountain Championship. If you're a fan and lucky enough you might still see one in action. In 1996, for instance, I saw a Lancia Delta S4 racing in Switzerland for the European Hill Climbing Championship, what a thrill!
To give you an idea of the kind of performance GroupB cars were capable of I'll mention that in the 1986 season Henri Toivonen made two laps around the Estoril circuit, during a stage of the Portuguese rally, the fastest of which, in 1 minute and 18,1 seconds, would have qualified him in the sixth position of the F1 Grand Prix that same season. Ayrton Senna had the Pole Position in the 1986 Portuguese Grand Prix in 1 minute and 16,7 seconds...Toivonen was using the Lancia Delta S4 and was accompanied by his usual co-driver Sergio Cresto. Keep in mind however that current Group A and WRC cars are even faster, overall, than GroupB cars used to be. This is mainly due to technology advances in tire formulations and suspension technology leading to Group A cars being faster around corners but losing on straights as compared to GroupB cars.
Note also that the engine used in the Metro 6R4, a 3.5 lt. V6, was used to power the famous Jaguar XJ220 of the early 90s (it was fitted with twin turbos in the XJ220).