= Solution
The <Mackey restriction formula> says that for subgroups $H,K\leq G$ and a $K$-representation $W$,
$$
\operatorname{Res}_H^G\operatorname{Ind}_K^GW
\cong
\bigoplus_{x\in H\backslash G/K}
\operatorname{Ind}_{H\cap xKx^{-1}}^H
\operatorname{Res}_{H\cap xKx^{-1}}^{xKx^{-1}}({}^xW),
$$
where ${}^xW(xkx^{-1})=W(k)$.
For $H=K=B$, the <Bruhat decomposition of SL2 over a finite field> gives
$$
G=B\sqcup BwB,
\qquad
w=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.
$$
Thus $B\backslash G/B$ has representatives $1,w$. The identity double coset contributes $\theta$, while
$$
B\cap wBw^{-1}=T
$$
contributes an induced representation from the diagonal subgroup. Therefore
$$
\boxed{
\operatorname{Res}_B^G\operatorname{Ind}_B^G\theta
\cong
\theta\oplus\operatorname{Ind}_T^B(\theta^w|_T),
}
$$
where
$$
\theta^w(t)=\theta(w^{-1}tw).
$$
For $t(a)=\operatorname{diag}(a,a^{-1})$, one has $w^{-1}t(a)w=t(a^{-1})$.
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