Let $\overline{a}$ and $\overline{c}$ are unit vectors and $\left|\overline{b}\right|=4$, the angle between $\overline{a}$ and $\overline{c}$ is ${\cos }^{-1}\left(\frac{1}{4}\right)$ and $\overline{b}-2\overline{C}=\lambda \overline{a}$ then the value of $\lambda$ is ____

Given $\overrightarrow{a}$ and $\overrightarrow{c}$ are any two unit vectors. 

Given $\left|\overrightarrow{b}\right|=4$ and $\cos \left(\overrightarrow{a},\overrightarrow{c}\right)=\frac{1}{4}$

Given $\overline{b}-2\overline{C}=\lambda \overline{a}$

This can be written as 

$\left|\overline{b}\right|=\left|\lambda \overline{a}+2\overline{c}\right|$

Squaring on both sides and substitute the appropraite values 

$$\begin{gathered} |\overline{b}{|}^2={\lambda }^2|\overline{a}{|}^2+4|\overline{c}{|}^2+4\lambda |\overline{a}\left|\right|\overline{c}|\cos \theta \\ 16={\lambda }^2(1)+4(1)+4\lambda (1\left)\right(1)\frac{1}{4} \end{gathered}$$

This can be written as 

${\lambda }^2+\lambda -12=0\Rightarrow \lambda =\boxed{\boxed{3,-4}}$