In ABGHR Thom spectra are described in the following way: we start with a morphism of Kan complexes $X\to \mathbb{S}\text{-line}$, where $\mathbb{S}\text{-line}$ is an $\infty$-groupoid which is equivalent to the well known $BGL_1(\mathbb{S})$. Then the Thom spectrum $Mf$ is defined to be the colimit of the composition of functors (of $\infty$-categories) $X\to\mathbb{S}\text{-line}\to\mathbb{S}Mod$. If we have a commutative ring spectrum $R$ there is a functor $\mathbb{S}\text{-line}\to R\text{-line}$ and it is a theorem that the colimit of the composition $X\to\mathbb{S}\text{-line}\to R\text{-line}\to RMod$ is equivalent to $Mf\wedge R$. There is a fibration of $\infty$-groupoids $R\text{-triv}\to R\text{-line}$ that we should think of as being our model of $EGL_1(R)\to BGL_1(R)$. If the composition $X\to R\text{-line}$ lifts to $R\text{-triv}$ then we say that $Mf$ is $R$-oriented and we have a Thom isomorphism $Mf\wedge R\simeq X_+\wedge R$.

My question is the following: we know classically that $Mf\wedge Mf\simeq X_+\wedge Mf $ (perhaps there are some conditions I need to put on $f$ to make this true?). So, if we assume $X$ is an infinite loop space and $X\to \mathbb{S}\text{-line}$ is a morphism of infinite loop spaces then we should have that $X\to\mathbb{S}\text{-line}\to Mf\text{-line}$ lifts to $Mf\text{-triv}$, yielding the classical Thom isomorphism described above. Is there an easy way to see that there is such a lift in this framework?

A more general question is: we know that such lifts always give such Thom isomorphisms, but are there conditions under which we can deduce a lift from such an isomorphism?