Kumari, who is a Member of Legislative Assembly (MLA) from Dalhousie in Himachal Pradesh, was given one-year imprisonment by a Chamba court in February this year but is currently on bail. Assembly elections are due in Punjab next year.
Nath, a veteran Congress leader, had to quit as All India Congress Committee (AICC) incharge of Punjab within three days after his appointment this month in the wake of an uproar by Bharatiya Janata Party (BJP), Shiromani Akali Dal (SAD) and Aam Aadmi Party (AAP) over his alleged role in the 1984 anti-Sikh riots.
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Kumari has served in various capacities in the party.
Party sources had earlier said that in Punjab the leadership could make a temporary appointment of an existing general secretary till a decision is taken on induction of new leaders in the AICC secretariat.
Kumari was convicted on February 26 in a land grab case by the Chamba court and sentenced to one year imprisonment, besides being fined Rs 8,000. The court had held her guilty of criminal conspiracy. On March 19, the Himachal Pradesh High Court suspended the one-year sentence.
An AICC secretary and four-time MLA, Kumari was facing trial in the case for over a decade along with six other persons.
Punjab BJP leader Vineet Joshi said the appointment was unfortunate and disgusting.
"First it was Kamal Nath and now Asha Kumari, who has been sentenced for 1-year jail by court in a case of land grab that too recently. I think they don't have good leaders," he said.
Nath had relinquished on June 15 from the charge of Punjab after BJP, SAD and AAP stepped up their attack on the party over an appointment.
He has refuted allegations against him.
Nath has said that he was hurt by developments wherein an unnecessary controversy has been created around the tragic 1984 riot.
The riots had followed the assassination of Indira Gandhi.