Many underwater acoustic (UWA) channels exhibit impulsive noise, thereby severely degrading the performance of traditional channel estimation algorithms. This paper presents two channel estimation algorithms for impulsive noise, namely (i) the variable forgetting factor l1,0 recursive least sign algorithm (VFF-l1,0-RLSA) and (ii) the variable forgetting factor l2,0 recursive least sign algorithm (VFF-l2,0-RLSA), both of which exploit the group sparse multipath structure and maintain robustness under impulsive noise. By using the l1 norm of the estimation error as part of the cost function, RLSAs are better at detecting and rejecting impulsive noise than the recursive least squares algorithms. A mixed l1,0 or l2,0 norm is incorporated with a RLSA to achieve better performance in group sparse UWA channel estimation. The time-varying forgetting factor and regularization parameter in the two proposed algorithms help to improve their performance. Simulation results based on Arctic ice cracking noise demonstrate the robustness and superiority of the two proposed algorithms.