MILITANT tendency - a Trotskyite group that set out to seize control of the British Labour Party a decade ago - is facing the likelihood of extinction in what promises to be a watershed by-election in the city of Liverpool.The July 4 contest for the parliamentary seat of Walton is widely seen as a make-or-break clash between the official British Labour Party, now under moderate leadership, and extreme-left socialists who claim to be the true representatives of Britain's working class. Unofficial polls conducted by Labour Party workers in the area suggest that Walton will probably be held by Peter Kilfoyle, the official Labour candidate. But interest centers on the performance of Lesley Mahmoud, who is campaigning as candidate of a splinter party calling itself "Real Labour," with Militant support. Mr. Kilfoyle has the backing of the Labour Party national leader, Neil Kinnock. Mrs. Mahmoud has described Mr. Kinnock's policies as a "sellout to the Conservatives." A sound defeat for Mahmoud is likely to be seen as the death-knell of the "hard left" in British politics. Kinnock would then be able to claim a highly symbolic victory, for the Militant Tendency was born in Walton in the mid-1960s. Kinnock began waging a campaign against Militant Tendency six years ago. By that time, the radical group had won control of the Liverpool City Council. Using its success there as a model, Militant tried to take charge of local Labour parties in other parts of Britain. It exploited the failure of moderate rank-and-file members to vote in elections for local party committees. Militant planned to engineer the selection around Britain of radical candidates to contest parliamentary elections for Labour. The ultimate aim was to build up a radical Labour Party majority in the House of Commons that could then form a government. As it pursued its takeover strategy in the 1970s, Militant regarded left-wing member of Parliament, Tony Benn, as its champion at Westminster. In 1980 Mr. Benn came within a few votes of being elected Labour's deputy leader. Fearing a Trotskyite takeover of the party, Kinnock set out to head off Militant nationally by invoking rules which forbid the existence of a party within the Labour Party. The strategy worked in Britain as a whole, but in Liverpool, despite a series of expulsions by the official Labour Party of local Militant members, the group has lived on. The reason lies in Liverpool's steep unemployment levels. In Walton, with its rows of drab red-brick terraced houses, 16 percent are jobless - twice the national average. Liverpool's decline from being one of the world's great port cities has resulted in much bitterness in a community suffering acutely from urban decay. Militant tried to exploit workers' sense of grievance. The expulsions of key Militant members from the Labour Party meant that a moderate Labour group was able to take over in the late 1980s. By then the city council had run up a debt of nearly British pounds1 billion pounds ($1.6 billion). The current by-election campaign happens to coincide with a bruising battle between the council and trade unions representing council refuse workers. The council ordered a series of job cuts in its over-staffed refuse department. A retaliatory overtime ban ordered by the trade unions produced thousands of tons of uncollected garbage in Liverpool's streets. Mahmoud and "Real Labour" sided with the unions and blamed the mountains of garbage on the council. Her Labour opponent backed the decision to order economies in the city administration. Nine days before polling day the garbage workers agreed to the job cuts. A French firm of private contractors will now clean Liverpool's streets. The garbage crisis sharpened an already acrimonious battle. At one point the trash collectors refused to collect garbage from houses displaying a Kilfoyle poster.