BEIRUT — Syria’s increasingly powerful Islamist rebel factions rejected the country’s new Western-backed opposition coalition and unilaterally declared an Islamic state in the key battleground of Aleppo, a sign of the seemingly intractable splits among those fighting to topple President Bashar Assad.

The move highlights the struggle over the direction of the rebellion at a time when the opposition is trying to gain the West’s trust and secure a flow of weapons to fight the regime. The rising profile of the extremist faction among the rebels could doom those efforts.