general relativity - Variation of the Christoffel symbols with respect to $g^{munu}$

I'm trying to find the field equations for some particular Lagrangian. In the middle I faced the term

$$\frac{\delta \Gamma_{\beta\gamma}^{\alpha}}{\delta g^{\mu\nu}} \, .$$

I know that

$$\delta \Gamma_{\beta\gamma}^{\alpha} = \frac{1}{2}g^{\sigma\alpha}(\nabla_{\beta}(\delta g_{\sigma\gamma}) + \nabla_{\gamma}(\delta g_{\sigma\beta}) - \nabla_{\sigma}(\delta g_{\beta\gamma})) \, .$$

I have two questions:

1. 
   

   Is the expression for $\delta \Gamma_{\beta\gamma}^{\alpha}$ somehow related to $\frac{\delta \Gamma_{\beta\gamma}^{\alpha}}{\delta g^{\mu\nu}}$?

   
   



2. 
   

   The idea at the end is to have terms like

   $$\frac{\delta \mathcal{L}}{\delta g^{\alpha\beta}} = 0$$ and thus make the variation of the action invariant under $\delta g^{\alpha\beta}$. So in simple words, is there is any way to have the term $\delta g^{\alpha\beta}$ put of the variation of the Christoffel symbol?