Pontryagin square, Postnikov square and their consistency formulas

1. $$\mathcal{P}_2$$ is Pontryagin square $$H^{2i}(M,\mathbb Z_{2^k})\to H^{4i}(M,\mathbb{Z}_{2^{k+1}}).$$

2. $$\mathfrak{P}$$ is the Postnikov square $$H^2(M,\mathbb Z_3)\to H^5(M,\mathbb Z_9).$$

We know for $$\mathbb Z_2$$-valued cocycles $$z_n$$, $$Sq^{n-k}(z_n) \equiv z_n\cup_{k} z_n$$ is always a cocycle. Here $$Sq$$ is called the Steenrod square.
More generally $$h_n \cup_{k} h_n$$ is a cocycle if $$n+k =$$ odd and $$h_n$$ is a cocycle.

If we define a generalized Steenrod square for cochains $$c_n$$: $$\tilde Sq^{n-k} c_n \equiv c_n\cup_{k} c_n + c_n\cup_{k+1} d c_n .$$ We can check $$d \tilde Sq^{k} c_n = d( c_n\cup_{n-k} c_n + c_n\cup_{n-k+1} d c_n )$$ $$= \tilde Sq^k d c_n, \;\;\;\; k=\text{odd}$$ $$=\tilde Sq^k d c_n +(-)^{n} 2 \tilde Sq^{k+1} c_n , \;\;\;\; k=\text{odd}.$$ This $$\tilde Sq^{2} c_2 \equiv c_2\cup_{0} c_2 + c_2\cup_{1} d c_2$$ almost is the same as the Pontryagin square $$\mathcal{P}_2$$ above, for ($$i=1,k=2$$ above) $$H^{2}(M,\mathbb Z_{2})\to H^{4}(M,\mathbb{Z}_{4}),$$ for $$\mathcal{P}_2 (x_2) \equiv x_2\cup_{0} x_2 + x_2\cup_{1} d x_2.$$

Are these generalized Steenrod squares known? ($$\tilde Sq^{n-k} c_n$$) Where can I find more discussions along this?

Comments about question (ii) For example, for Steenrod square, the total Stiefel-Whitney class $$w=1+w_1+w_2+\cdots$$ is related to the total Wu class $$u=1+u_1+u_2+\cdots$$ through the total Steenrod square $$w=Sq(u),\ \ \ Sq=1+Sq^1+Sq^2+ \cdots .$$ Therefore, $$w_n=\sum_{i=0}^n Sq^i (u_{n-i})$$. The Steenrod squares have the following properties: $$Sq^i(x_j) =0, \ i>j, \ \ Sq^j(x_j) =x_jx_j, \ \ Sq^0=1,$$

Do we have something similar for thse "Bockstein homomorphism?" $$\beta_p$$, $$\beta_p'$$, $$\beta_{2^n}$$?

Massey proves an analogue for the Pontryagin square of Thom's formula $$w_k=\Phi^{-1}Sq^k(u)$$ for the Stiefel-Whitney classes, which seems relevant to your question (ii).