According to Magma it is 120, and it is an extension of $A_5$ by $C_2$
( A:=AutomorphismGroup(HyperellipticCurve(Polynomial(GF(5),[0,-1,0,0,0,1]))); )
And over $F_{25}$ or over $\bar F_5$ it is 240.
Tim Dokchitser
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