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Be careful that your lifted polynomial is not a polynomial in ZZ but in GF(q). You can check with

sage: aa.lift().parent()                                                                                                                                                                                            
Univariate Polynomial Ring in x over Finite Field of size 7

What you actually want is the "centered lift of this lift". A one line solution is

sage: ZZ['x']([coeff.lift_centered() for coeff in aa.lift()])                                                                                                                                                       
-3*x^4 + 2*x^3 + x^2 + 3*x - 2

Not very elegant but it does the job.