<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0609</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2026-09-28</periodOfReport>

    <issuer>
        <issuerCik>0001422143</issuerCik>
        <issuerName>Kura Oncology, Inc.</issuerName>
        <issuerTradingSymbol>KURA</issuerTradingSymbol>
        <issuerForeignTradingSymbol></issuerForeignTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001910331</rptOwnerCik>
            <rptOwnerName>DOYLE THOMAS JAMES</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerNonUSAddressFlag>false</rptOwnerNonUSAddressFlag>
            <rptOwnerStreet1>C/O KURA ONCOLOGY, INC.</rptOwnerStreet1>
            <rptOwnerStreet2>4930 DIRECTORS PLACE, SUITE 500</rptOwnerStreet2>
            <rptOwnerCity>SAN DIEGO</rptOwnerCity>
            <rptOwnerState>CA</rptOwnerState>
            <rptOwnerZipCode>92121</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>false</isDirector>
            <isOfficer>true</isOfficer>
            <isTenPercentOwner>false</isTenPercentOwner>
            <isOther>false</isOther>
            <officerTitle>SVP, Finance &amp; Accounting</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <aff10b5One>false</aff10b5One>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2026-09-28</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>false</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>4607</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>10.4972</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>142428</value>
                    <footnoteId id="F2"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>500</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>spouse's 401(k)</value>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
    </nonDerivativeTable>

    <footnotes>
        <footnote id="F1">Sell-to-cover for taxes associated with the vesting of 1/6th of the underlying shares of performance-based restricted stock units (&quot;PSUs&quot;) granted to the Reporting Person on May 31, 2023.  Each PSU represents the contingent right to receive one share of the Issuer's common stock based on the achievement of each of three specified development milestones, and the one-year anniversary of each milestone achievement, subject to the Reporting Person's continuous service on each corresponding vesting date. The one-year anniversary of the specified development milestone was determined to be met on September 27, 2026, resulting in the vesting of 1/6th of the underlying shares.</footnote>
        <footnote id="F2">Includes 1,868 shares acquired on May 17, 2026 pursuant to the Issuer's Employee Stock Purchase Plan.</footnote>
    </footnotes>

    <ownerSignature>
        <signatureName>Teresa Bair, Attorney-in-fact for Thomas Doyle</signatureName>
        <signatureDate>2026-09-29</signatureDate>
    </ownerSignature>
</ownershipDocument>
