Perfect aggregation of information in multistage two-person games with a fixed order of moves and aggregate information about the partner's choice

Two-person multistage game is considered with fixed order of moves, with perfect information at every move about the history of the game, and with aggregated information about current move of the second player. Player 1 is the first to choose his move $x_i(\cdot)$ on every stage and at the beginning of the game announces his strategy $x(\cdot)=(x_1(\cdot),\dots,x_n(\cdot))$ – the complete plan for all $n$ stages. Given the choice of player 1 and the history of the game the second player maximizes her payoff function by choosing her strategy $\nu=(\nu_1,\nu_2,\dots,\nu_n)$. The article applies the theory of Lee groups to give sufficient condition of perfect information aggregation in this game.