Crossed module structure on $\pi_1$-level of any map $f: X\to Y$ For cofibration $f:A\to X$ we have crossed module $\pi_2(X,A)\to\pi_1(A)$. On other hand, we can change map $f$ to fibration and consider crossed module 
$\pi_1(E)\to\pi_1(P_f)$, where $P_f$ is relative path space and $E$ is homotopy fiber of $f$. Homotopy exact sequence of pair and long exact sequence associated with $E\to P_f\to X$ are essentially the same and judging by action of $\pi_1(A)$ (resp. $\pi_1(P_f)$) these crossed modules are isomorphic. So, looks like isomorphism $\pi_1(A)\cong \pi_1(P_f)$ preserve crossed module structure.
Question
Is it true, that for any map of spaces $f:X\to Y$, we can form a crossed module $\pi_1(\mathrm{hofib}(f))\to\pi_1(X)$ using isomorphism $\pi_1(X)\cong \pi_1(P_f)$ ?
 A: First of all, you should make quite clear that you are working in the category of spaces with base point. 
Secondly, for any map $f: A \to X$ of pointed spaces the map $\pi_1(F(f)) \to \pi_1(X)$, where $F(f)$ is the homotopy fibre of $f$, may be given the structure of a crossed module. 
There are several ways of proving the trickier part,  the second rule for a crossed module, which is that
$$v^{-1}uv=u^a, \text{where } a=\partial v.$$  I prefer emphasis on the $2$-dimensional approach and the interchange law as suggested by the following diagram:

which is taken from Proposition 6.2.4 of Nonabelian Algebraic Topology, for which a pdf is available there. In this picture, $u,v$ can be taken to represent elements of a second relative homotopy  group, the square figure is a constant map, and the vertical pairs represent vertical identities. If you add the columns first and then the rows, you get the left hand side; if you add the rows first and then the columns, you get the right hand side. 
I mention that the paper given here has a generalisation of this result, giving a homotopy double groupoid of any map of spaces, not pointed. 
