**Notations:** Let $K$ be a locally compact Hausdorff space and $E$ be a real normed linear space.
Recall that $C_0(K,E)$ is the set of $E$-valued continuous functions $f$ on $K$ such that $f$ vanishes at infinity.
For any $t\in K,$ let $\psi_t$ be an evaluation functional on $C_0(K,E).$
If $X^*$ is a dual space of $X,$ then denote $ext(X^*)$ to be the set of extreme points of the unit ball of $X^*.$

The following corollary is quoted in the book ['Isometries on Banach Spaces: Function Spaces' by Fleming and Jamison][1], Chapter $2,$ page $33,$ 

> Corollary $2.3.6.$ If $X$ is a subspace of $C_0(K,E),$ then 
$$ext(X^*) \subset \{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}.$$ 

I would like to ask whether the reverse inclusion is true or not, that is, 
does 
$$\{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}\subset ext(X^*)$$
hold? 
If yes, can provide a proof or its reference? 
Otherwise, can I have counterexample?

  [1]: https://www.crcpress.com/Isometries-on-Banach-Spaces-function-spaces/Fleming-Jamison/p/book/9781584880400