Let $\mathcal{M}[0,1]$ be the $\sigma$-algebra of Lebesgue-measurable subsets of $[0,1]$ and $\lambda$ be the Lebesgue measure.
Are there two non-Lebesgue-measurable subsets $E_1$ and $ E_2$ of $[0,1]$ ($\not \in \mathcal{M}[0,1])$ and a countably-additive extension $\mu$ of $\lambda$ to the $\sigma$-algebra $\sigma(\mathcal{M}[0,1])\cup \{E_1, E_2\})$ such that
(1) $(\mu\upharpoonright_{\mathcal{M}_1})_* (E_2) \neq (\mu\upharpoonright_{\mathcal{M}_1})^* (E_2)$
(2) $(\mu\upharpoonright_{\mathcal{M}_2})_* (E_1) \neq (\mu\upharpoonright_{\mathcal{M}_2})^* (E_1)$
where
$\mathcal{M}_1=\sigma(\mathcal{M}[0,1]\cup \{E_1\})$ and $\mathcal{M}_2=\sigma(\mathcal{M}[0,1]\cup \{E_2\})$?