Socialist caretaker Prime Minister Bettino Craxi was asked today to try to form a new government, 24 days after his resignation over a minor parliamentary defeat plunged Italy into a political crisis.

President Francesco Cossiga's request came after Christian Democrat Giulio Andreotti, Craxi's foreign minister for almost three years, reported failure of his 10-day effort to form a government because of a lack of Socialist support.

While Craxi, 52, promised to try to reconstruct his five-party coalition, a spokesman for the Christian Democrats, the largest party in the old coalition, predicted failure. "Craxi will now find himself before the same difficulty that Andreotti faced," said Clemente Mastella.

Andreotti failed because the Socialists refused to accept a Christian Democrat as premier on the grounds that the Christian Democrats had arbitrarily brought down Craxi's government -- the longest lasting since the end of World War II.

Christian Democratic Party leader Ciriaco de Mita has demanded that Craxi accept the principle of alternating the prime ministership between their parties, the two largest in the coalition, as a condition for further support.

Craxi has refused to make any deal that would limit the duration of his rule, on the bases that such a limit should not be determined unilaterally by the Christian Democrats, and that he would not be able to govern effectively as a lame duck.