May 20 (Reuters) - FirstGroup's second-biggest shareholder, Schroders, said on Thursday it planned to vote against the company's $4.6 billion sale of its North American bus businesses to private equity firm EQT, joining top investor Coast Capital in opposing the deal.

The move comes after activist hedge fund Coast Capital asked other FirstGroup investors to vote against the proposed sale of the "crown jewel assets" unless the offer was improved.

"We appreciate this has been a very difficult period in which to carry out a sales process," a Schroders spokesperson said in an emailed statement.

"However, after careful consideration of the merits of the proposed sale...we have the intention of voting against the deal as it stands, as we believe it undervalues these businesses."

FirstGroup said on Tuesday it "followed a comprehensive and competitive process in order to seek the best possible price" for the assets.

The deal has the backing of Columbia Threadneedle Investments, FirstGroup's third-largest shareholder with a stake of around 10%.

Schroder Investment Management owns nearly 12% of FirstGroup, while Coast Capital owns nearly 14%, Refinitiv Eikon data showed.

"We continue to have full confidence in Chairman David Martin to provide the necessary leadership and vision for the ongoing business," the spokesperson for Schroders added.

Coast Capital founding partner James Rasteh welcomed Schroders' move to vote against the deal but declined to comment on whether the fund still has confidence in Martin.

FirstGroup plans to use the proceeds to pay down debt, contribute to its UK pension schemes and return money to shareholders.

Coast, however, also criticized the portion of proceeds that shareholders stand to receive from the deal.

FirstGroup, which will focus on its UK bus and train operations and the sale of Greyhound, had said that it would consider making additional distributions to continuing shareholders on top of the 365 million pounds it had earmarked.

The company will hold a general meeting on May 27 to vote on the proposed deal.

(Reporting by Yadarisa Shabong in Bengaluru; Editing by Aditya Soni, Kirsten Donovan)